New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems

Saima Rashid; Saima Parveen; Hijaz Ahmad; Yu-Ming Chu

doi:10.1515/phys-2021-0001

Open Access Published by De Gruyter Open Access February 25, 2021

New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems

Saima Rashid , Saima Parveen , Hijaz Ahmad and Yu-Ming Chu

From the journal Open Physics

https://doi.org/10.1515/phys-2021-0001

Abstract

In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard ( ℋℋ )-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the ℋℋ type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.

Keywords: quantum estimates; Hermite–Hadamard inequalities; convex functions; generalized ψ-convex functions; Raina function

1 Introduction

Let ℐ be an interval in R . Then G : ℐ ↦ R is said to be convex if

G ( ξ x + ( 1 − ξ ) y ) ≤ ξ G ( x ) + ( 1 − ξ ) G ( y )

holds for all x , y ∈ ℐ and ξ ∈ [0, 1].

Convex functions have potential applications in many intriguing and captivating fields of research and furthermore played a remarkable role in numerous areas, such as coding theory, optimization, physics, information theory, engineering and inequality theory. Several new classes of classical convexity have been proposed in the literature, see refs [1,2]. Many researchers endeavored, attempted and maintained their work on the concept of convex functions and generalized its variant forms in different ways using innovative ideas and fruitful techniques [3,4]. Many mathematicians always kept continually hardworkingin the field of inequalities and have collaborated with different ideas and concepts in the theory of inequalities and its applications, see refs [5,6, 7,8,9, 10,11,12, 13,14]. Many inequalities are proved for convex functions, but the most known from the related literature is the Hermite–Hadamard inequality.

Let G : ℐ ⊆ R ↦ R be a convex function such that η ₁ < η ₂. Then

(1.1) G η 1 + η 2 2 ≤ 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d z ≤ G ( η 1 ) + G ( η 2 ) 2 .

The inequality (1.1) is a well-known paramount in the related literature and plays its pivotal role in optimization, coding and fractional calculus theory [15,16,17, 18,19,20, 21,22,23, 24,25].

Many studies have recently been carried out in the field of q-analysis [26,27,28, 29,30,31, 32,33,34,39], starting with Euler owing to an extraordinary demand for mathematics that models quantum figuring q-calculus performed as an association between mathematics and physics. Several mathematical areas have been correlated with quantum calculus such as fractional diffusion equations, special theory of relativity, quantum mechanics, orthogonal polynomials and henceforth. The mathematical description of a quantum system typically takes the form of a “wavefunction,” generally represented in equations by the Greek letter psi: ψ. Apparently, Euler was the founder of this branch of mathematics, by using the parameter q in Newton’s work of infinite series. Later, Jackson was the first to develop q-calculus that is known without limits calculus in a systematic way [36]. In 1908–1909, Jackson defined the general q-integral and q-difference operator [35]. In 1969, Agarwal described the q-fractional derivative for the first time [37]. In 1966–1967, Al-Salam introduced q-analogs of the Riemann–Liouville fractional integral operator and q-fractional integral operator [38]. In 2004, Rajkovic gave a definition of the Riemann-type q-integral which was the generalization of Jackson q-integral. In 2013, Tariboon introduced D q η 1 -difference operator [42].

Inspired by the aforementioned literature on the improvement of the correlation of quantum calculus and convexity theory, we addressed the notion of generalized ψ-s-convex functions and generalized ψ-quasi-convex functions. Taking into consideration, a q-integral identity, we derived some new estimates of Hermite–Hadamard inequalities for twice differentiable functions via the aforesaid classes of generalized ψ-convex functions. Relevant connections of the several consequences demonstrated here with those associating relatively some well-known classical convex functions are also apprehended.

2 Preliminaries

First, suppose there is an arbitrary non-negative function ℏ : ( 0 , 1 ) → R , ϑ = { ϑ ( m ) } m = 0 ∞ be a bounded sequence of real numbers and ℱ υ 1 , υ 2 ϑ ( . ) υ 1 , υ 2 > 0 denotes Raina’s function.

In ref. [40], R. K. Raina explored a new class of functions stated as:

(2.1) ℱ υ 1 , υ 2 ϑ ( z ) = ℱ υ 1 , υ 2 ϑ ( 0 ) , ϑ ( 1 ) , … ( z ) = ∑ m = 0 ∞ ϑ ( m ) Γ ( υ 1 m + ϑ ) z m ,

where υ 1 , υ 2 > 0 , ∣ z ∣ < R and

ϑ = ( ϑ ( 0 ) , … , ϑ ( m ) , … )

is a bounded sequence of positive real numbers. Note that if we choose υ ₁ = 1, υ ₂ = 0 in (2.1), then

ϑ ( m ) = ( δ 1 ) m ( δ 2 ) m ( δ 3 ) m f o r m = 0 , 1 , 2 , … ,

where δ ₁, δ ₂ and δ ₃ are parameters which can choose arbitrary real and complex values (provided that δ ₃ ≠ 0, −1, −2, …,) and we have the notion (b)_m by

( b ) m = Γ ( b + m ) Γ ( b ) = b ( b + 1 ) … ( b + m − 1 ) , m = 0 , 1 , 2 , … ,

then the classical hypergeometric function is stated as follows:

ℱ υ 1 , υ 2 ϑ ( z ) = F ( δ 1 , δ 2 ; δ 3 ; z ) = ∑ m = 0 ∞ ( δ 1 ) m ( δ 2 ) m m ! ( δ 3 ) m z m , ∣ z ∣ ≤ 1 , z ∈ C .

Also, if ϑ = (1, 1,…) with ς = δ, (ℜ(δ) > 0), ϑ = 1 and restricting its domain to z ∈ C in (2.1), then we have the classical Mittag-Leffler function:

E δ 1 ( z ) = ∑ m = 0 ∞ 1 Γ ( 1 + δ 1 m ) z κ .

Next, we evoke a novel concept of set and mappings including Raina’s functions.

Definition 2.1

[41] A non-empty set K ℱ is said to be a generalized ψ-convex set, if

(2.2) η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∈ K ℱ

for all η 1 , η 2 ∈ K ℱ , ξ ∈ [ 0 , 1 ] .

We now define the generalized ψ-convex function presented by Vivas-Cortez et al. [41].

Definition 2.2

[41] Let a set K ℱ ⊆ R and a mapping G : K ℱ → R is said to be generalized ψ-convex, if

(2.3) G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) ≤ ( 1 − ξ ) G ( η 1 ) + ξ G ( η 2 ) for all η 1 , η 2 ∈ K ℱ , ξ ∈ [ 0 , 1 ] .

Next, we present another idea of generalized ψ-convex functions for an arbitrary nonnegative function ℏ.

Definition 2.3

Let ℏ : ( 0 , 1 ) → R be a real mapping and G : K ℱ → R is said to be a generalized (ψ, ℏ)-convex function, if

(2.4) G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) ≤ ℏ ( 1 − ξ ) G ( η 1 ) + ℏ ( ξ ) G ( η 2 ) for all η 1 , η 2 ∈ K ℱ , ξ ∈ [ 0 , 1 ] .

Furthermore, we demonstrate a new class of generalized ψ-convex functions with respect to an arbitrary non-negative function is known as the generalized ψ-s-convex function.

Definition 2.4

Let s ∈ (0, 1] and a mapping G : K ℱ → R is said to be generalized ψ-s-convex, if

(2.5) G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) ≤ ( 1 − ξ ) s G ( η 1 ) + ξ s G ( η 2 ) for all η 1 , η 2 ∈ K ℱ , ξ ∈ [ 0 , 1 ] .

Definition 2.5

Let a function G : K ℱ → R is said to be generalized ψ-quasi-convex, if

(2.6) G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) ≤ sup G ( η 1 ) , G ( η 2 ) for all η 1 , η 2 ∈ K ℱ , ξ ∈ [ 0 , 1 ] .

It is obvious that any generalized ψ-convex function is a generalized ψ-quasi-convex function but converse may not be true.

In this section, we first evoke certain earlier famous notions on q-calculus that will be helpful throughout the investigation.

Consider an interval J = [ η 1 , η 2 ] ⊆ R and 0 < q < 1 and be a constant.

Definition 2.6

[42] Let a continuous mapping G : J ↦ R and suppose z ∈ J . Then q-derivative on J of function G at z is stated as

(2.7) D q η 1 G ( z ) = G ( z ) − G ( q z + ( 1 − q ) η 1 ) ( 1 − q ) ( z − η 1 ) , z ≠ η 1 , D q η 1 G ( η 1 ) = lim z ↦ η 1 D q η 1 G ( z ) .

We say that G is q-differentiable on J provided D q η 1 G ( z ) exists for all z ∈ J . Note that if η ₁ = 0 in (2.7), then D q 0 G ( z ) = D q G , where D q is the worthmentioning q-derivative of the mapping G ( z ) stated by

(2.8) D q G ( z ) = G ( z ) − G ( q z ) ( 1 − q ) z .

Definition 2.7

[42] Let a continuous mapping G : J ↦ R and suppose the second-order q-derivative on interval J , which is identified as D q 2 η 1 G , provided D q 2 η 1 G is q-differentiable on J with D q 2 η 1 G = D η 1 ( D q η 1 G ) : J ↦ R . Analogously, we present higher order q-derivative on J , D q n η 1 : J κ ↦ R .

Definition 2.8

[42] Let a continuous mapping G : J ⊂ R ↦ R and its q-integral on J is presented as

(2.9) ∫ η 1 z G ( ξ ) η 1 d q 1 ξ = ( 1 − q ) ( z − η 1 ) ∑ n = 0 ∞ q n G ( q n z + ( 1 − q n ) η 1 )

for z ∈ J . Also, if c ₁ ∈ (η ₁, z), then the definite q-integral on J is stated as follows:

∫ c 1 z G ( ξ ) η 1 d q 1 ξ = ∫ η 1 z G ( ξ ) d q 1 η 1 ξ − ∫ η 1 c 1 G ( ξ ) d q 1 η 1 ξ = ( 1 − q ) ( z − η 1 ) ∑ n = 0 ∞ q n G ( q n z + ( 1 − q n ) η 1 ) − ( 1 − q ) ( c 1 − η 1 ) ∑ n = 0 ∞ q n G ( q n c 1 + ( 1 − q n ) η 1 ) ,

It is observed that if η ₁ = 0, then we have the classical q-integral, which is stated as

(2.10) ∫ 0 z G ( ξ ) d q 1 0 ξ = ( 1 − q ) z ∑ n = 0 ∞ q n G ( q n z ) for z ∈ [ 0 , ∞ ) .

Theorem 2.1

[42] Let two continuous functions G , g 1 : J ↦ R with c ∈ R . Then, for z ∈ J ,

∫ η 1 z [ G ( ξ ) + g 1 ( z ) ] η 1 d q 1 ξ = ∫ c 1 z G ( ξ ) d q 1 η 1 ξ + ∫ c 1 z g 1 ( ξ ) d q 1 η 1 ξ ; ∫ c 1 z ( c G ) ( ξ ) d q 1 η 1 ξ = c ∫ c 1 z G ( ξ ) d q 1 η 1 ξ .

Additionally, we propose the q-analogues of η ₁ and ( z − η 1 ) n and the concept of q-beta function.

Definition 2.9

[43] For any real number η ₁,

(2.11) [ η 1 ] = q n − 1 q − 1

is known as the q-analogue of η ₁. Specifically, if n ∈ Z + , we symbolize

[ n ] = q n − 1 q − 1 = q n − 1 + ⋯ + q + 1 .

Definition 2.10

[43] If n is an integer, the q-analogue of ( z − η 1 ) n is the polynomial

(2.12) ( z − η 1 ) q n = 1 , if n = 0 , ( z − η 1 ) ( z − q η 1 ) ⋯ ( z − q n − 1 η 1 ) , if n ≥ 1 .

Definition 2.11

For any ξ, ζ > 0,

(2.13) B q ( ξ , ζ ) = ∫ 0 1 z ξ − 1 ( 1 − q z ) q ζ − 1 d q 0 z

is called the q-beta function. It is observe that

(2.14) B q ( ξ , 1 ) = ∫ 0 1 z ξ − 1 d q 0 z = 1 [ ξ ] ,

where [ξ] is the q-analogue of ξ.

The succeeding lemmas will be needed in the proof of our theorems.

Lemma 2.2

Assume that G ( z ) = 1 , then

∫ 0 1 d q 0 z = ( 1 − q ) ∑ n = 0 ∞ q n = 1 .

Lemma 2.3

Assume that G ( z ) = z for z ∈ [η ₁, η ₂], then

∫ 0 1 z d q 0 z = ( 1 − q ) ∑ n = 0 ∞ q 2 n = 1 1 + q .

Lemma 2.4

Assume that G ( z ) = 1 − q z for z ∈ [η ₁, η ₂] and 0 < q < 1 be a constant, then

∫ 0 1 ( 1 − q z ) d q 0 z = ∫ 0 1 d q 0 z − q ∫ 0 1 z d q 0 z = 1 1 + q .

Lemma 2.5

Assume that G ( z ) = z ( 1 − q z ) for z ∈ [η ₁, η ₂] and 0 < q < 1 be a constant, then

∫ 0 1 z ( 1 − q z ) d q 0 z = ∫ 0 1 z d q 0 z − q ∫ 0 1 z 2 d q 0 z = 1 1 + q − q ( 1 − q ) ∑ n = 0 ∞ q 3 n = 1 ( 1 + q ) ( 1 + q + q 2 ) .

In ref. [44], Vivas-Cortez et al. derived the following q-integral identity for generalized ψ-convex functions.

Lemma 2.6

[44] Let υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a continuous and twice q-differentiable function G : ϒ = [ υ 1 , υ 1 + ℱ υ 1 , υ 2 ϑ ( υ 2 − υ 1 ) ] ⊂ R → R on ϒ^∘ (the interior of ϒ) having ℱ υ 1 , υ 2 ϑ ( υ 2 − υ 1 ) > 0 such that D q 2 η 1 G is integrable on [ υ 1 , υ 1 + ℱ υ 1 , υ 2 ϑ ( υ 2 − υ 1 ) ] . Then the following equality holds:

(2.15) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) × ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q × ∫ 0 1 ξ ( 1 − q ξ ) D q 2 η 1 G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) d q 0 ξ .

Striving by the abovementioned work, the presentation of this paper is as follows: In Section 3, the ℋℋ -type variants for generalized ψ-s-convex functions are demonstrated by using new quantum integral identity. In Section 4, numerous novel q-estimates of ℋℋ -type variants for generalized ψ-quasi-convex functions for twice q-differentiable functions are generalized in detail. Taking these findings into account, we derive certain quantum bounds for the aforesaid functional classes. Remarkable special cases are established. A detailed conclusion with open problems is presented in Section 5.

3 Differentiable ℋℋ -type inequalities for generalized ψ-s-convex functions

The main purpose of this article is to establish some variants of ℋℋ -type inequalities for ψ-s-convex functions. In what follows, we use Lemma 2.6.

Theorem 3.1

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If ∣ D q 2 η 1 G ∣ α is a generalized ψ-s-convex function on ϒ for α ≥ 1, and α ⁻¹ + β ⁻¹ = 1, then

(3.1) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , β + 1 ) 1 β × [ s + 1 ] q − 2 s − 1 2 s − 1 D q 2 η 1 G ( η 1 ) α + D q 2 η 1 G ( η 2 ) α [ s + 1 ] q 1 / α .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-s-convex function with Hölder’s inequality and from Lemma 2.6, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ ( 1 − q ξ ) D q 2 η 1 ∣ G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) ∣ d q 0 ξ ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ β ( 1 − q ξ ) β d q 0 ξ 1 β × ∫ 0 1 D q 2 η 1 G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) α d q 0 ξ 1 / α ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , β + 1 ) 1 β × ∫ 0 1 ( 1 − ξ ) s D q 2 η 1 G ( η 1 ) α + ξ s D q 2 η 1 G ( η 2 ) α d q 0 ξ 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , β + 1 ) 1 β × [ s + 1 ] q − 2 s − 1 2 s − 1 D q 2 η 1 G ( η 1 ) α + D q 2 η 1 G ( η 2 ) α [ s + 1 ] q 1 / α .

This completes the proof of Theorem 3.5.□

Corollary 3.2

If in Theorem 3.5 letting D q 2 η 1 G ≤ ℳ , then we get

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ ℳ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , β + 1 ) 1 β × [ s + 1 ] q − 2 s − 1 2 s − 2 1 [ s + 1 ] q 1 / α .

Remark 3.1

Letting s = 1, then inequality (3.1) coincides with Theorem 6 in ref. [44].

Theorem 3.3

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If ∣ D q 2 η 1 G ∣ α is a generalized ψ-s-convex function on ϒ for α ≥ 1, and α ⁻¹ + β ⁻¹ = 1, then

(3.2) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 q + 1 1 − 1 α × A 1 ( q ; ξ ) D q 2 η 1 G ( η 1 ) α + A 2 ( q ; ξ ) D q 2 η 1 G ( η 2 ) α 1 / α ,

where

A 1 ( q ; ξ ) ≔ 2 1 − s B q ( α + 1 , 2 ) − B q ( s + 1 , α + 1 )

and

A 2 ( q ; ξ ) ≔ B q ( α + 1 , s + 2 ) .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-s-convex function with Hölder’s inequality and from Lemma 2.6, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ ( 1 − q ξ ) D q 2 η 1 × G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) d q 0 ξ ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ d q 0 ξ 1 − 1 α × ∫ 0 1 ξ ( 1 − q ξ ) α D q 2 η 1 G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) α d q 0 ξ 1 / α ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 q + 1 1 − 1 α × ∫ 0 1 ξ ( 1 − q ξ ) α ( 1 − ξ ) s D q 2 η 1 G ( η 1 ) α + ξ s D q 2 η 1 G ( η 2 ) α d q 0 ξ 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 q + 1 1 − 1 α × ∫ 0 1 ξ ( 1 − q ξ ) α ( 2 1 − s − ξ s ) D q 2 η 1 G ( η 1 ) α + ξ s D q 2 η 1 G ( η 2 ) α d q 0 ξ 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 q + 1 1 − 1 α × A 1 ( q ; ξ ) D q 2 η 1 G ( η 1 ) α + A 2 ( q ; ξ ) D q 2 η 1 G ( η 2 ) α 1 / α ,

where

A 1 ( q ; ξ ) ≔ ∫ 0 1 ξ ( 2 1 − s − ξ s ) ( 1 − q ξ ) α d q 0 ξ = 2 1 − s B q ( α + 1 , 2 ) − B q ( s + 1 , α + 1 )

and

A 2 ( q ; ξ ) ≔ ∫ 0 1 ξ s + 1 ( 1 − q ξ ) α d q 0 ξ = B q ( α + 1 , s + 2 ) .

This completes the proof of Theorem 3.3.□

Corollary 3.4

If in Theorem 3.3, letting D q 2 η 1 G ≤ ℳ , then we get

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ ℳ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 q + 1 1 − 1 α × A 1 ( q ; ξ ) + A 2 ( q ; ξ ) 1 / α .

Remark 3.2

Letting s = 1, then inequality (3.2) coincides with Theorem 11 in ref. [44].

Theorem 3.5

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-s-convex function on ϒ for α ≥ 1, and α ⁻¹ + β ⁻¹ = 1, then

(3.3) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q A 3 ( q ; ξ ) D q 2 η 1 G ( η 1 ) α + A 4 ( q ; ξ ) D q 2 η 1 G ( η 2 ) α 1 / α ,

where

A 3 ( q ; ξ ) ≔ 2 1 − s B q ( α + 1 , α + 1 ) − B q ( α + s + 1 , α + 1 )

and

A 4 ( q ; ξ ) ≔ B q ( α + s + 1 , α + 1 ) .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-s-convex function with Hölder’s inequality and from Lemma 2.6, we have

(3.4) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ ( 1 − q ξ ) D q 2 η 1 × G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) d q 0 ξ ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 d q 0 ξ 1 β × ∫ 0 1 ξ α ( 1 − q ξ ) α D q 2 η 1 G ( η 1 + ξ ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) α d q 0 ξ 1 / α ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 d q 0 ξ 1 β × ∫ 0 1 ξ α ( 1 − q ξ ) α ( 1 − ξ ) s D q 2 η 1 G ( η 1 ) α + ξ s D q 2 η 1 G ( η 2 ) α d q 0 ξ 1 / α .

Applying Lemma 2.5, we have

(3.5) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q A 3 ( q ; ξ ) D q 2 η 1 G ( η 1 ) α + A 4 ( q ; ξ ) D q 2 η 1 G ( η 2 ) α 1 / α ,

using the fact that

A 3 ( q ; ξ ) ≔ ∫ 0 1 ξ α ( 1 − q ξ ) α ( 1 − ξ ) s d q 0 ξ = 2 1 − s B q ( α + 1 , α + 1 ) − B q ( α + s + 1 , α + 1 )

and

A 4 ( q ; ξ ) ≔ ∫ 0 1 ξ α ( 1 − q ξ ) α ξ s d q 0 ξ = B q ( α + s + 1 , α + 1 ) .

This completes the proof of Theorem 3.5.□

4 ℋℋ -type inequalities for generalized ψ-quasi-convex functions

The main purpose of this article is to establish some variants of ℋℋ -type inequalities for ψ-convex functions. In what follows, we use Lemma 2.6.

Theorem 4.1

Let υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-quasi-convex function on ϒ for α ≥ 1, and α ⁻¹ + β ⁻¹ = 1, then

(4.1) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 1 + q 1 − 1 / α × Ω 1 ( q ; ξ ) sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

where

(4.2) Ω 1 ( q ; ξ ) ≔ ( 1 − q ) ∑ n = 0 ∞ q 2 n ( 1 − q n + 1 ) α .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

Applying Lemma 2.3, we have

(4.3) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 1 + q 1 − 1 / α × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α × ∫ 0 1 ξ ( 1 − q ξ ) α d q 0 ξ 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 1 + q 1 − 1 / α × Ω 1 ( q ; ξ ) sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

using the fact that

Ω 1 ( q ; ξ ) ≔ ∫ 0 1 ξ ( 1 − q ξ ) α d q 0 ξ = ( 1 − q ) ∑ n = 0 ∞ q 2 n ( 1 − q n + 1 ) r .

This completes the proof of Theorem 4.1.□

Corollary 4.2

If α is taken to be positive integer, then under the assumption of Theorem 4.1, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 1 + q 1 − 1 / α × B q ( α + 1 , 2 ) sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α .

Remark 4.1

If in Theorem 4.1 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1

G ( η 1 ) + G ( η 2 ) 2 − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d z ≤ ( η 2 − a 1 ) 2 4 2 ( α + 1 ) ( α + 2 ) 1 / α sup G ″ ( η 1 ) α , G ″ ( η 2 ) α 1 / α .

This coincides with Theorem 2 in ref. [45].

Theorem 4.3

(4.4) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 2 ( q ; ξ ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

where

Ω 2 ( q ; ξ ) ≔ ( 1 − q ) ∑ n = 0 ∞ q n ( β + 1 ) ( 1 − q n + 1 ) β .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

Applying Lemma 2.2, we have

(4.5) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 2 ( q ; ξ ) 1 / α × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

using the fact that

Ω 2 ( q ; ξ ) ≔ ∫ 0 1 ξ β ( 1 − q ξ ) β d q 0 ξ = ( 1 − q ) ∑ n = 0 ∞ q n ( β + 1 ) ( 1 − q n + 1 ) β .

This completes the proof of Theorem 4.3.□

Corollary 4.4

If β > 1 is taken to be positive integer, then under the assumption of Theorem 4.3, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , β + 1 ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α .

Remark 4.2

If in Theorem 4.1 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1

G ( η 1 ) + G ( η 2 ) 2 − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d z ≤ ( η 2 − η 1 ) 2 8 π 2 1 / β Γ ( 1 + β ) Γ ( 1.5 + β ) 1 / β × sup G ″ ( η 1 ) α , G ″ ( η 2 ) α 1 / α .

This coincides with the result in ref. [1].

Theorem 4.5

Let υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0), …, ϑ(κ), …) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-quasi-convex function on ϒ for α, β > 1, and α ⁻¹ + β ⁻¹ = 1, then

(4.6) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 3 ( q ; ξ ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 + q 1 / α ,

where

Ω 3 ( q ; ξ ) ≔ ( 1 − q ) ∑ n = 0 ∞ q 2 n ( 1 − q n + 1 ) β .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

Applying Lemma 2.3, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 3 ( q ; ξ ) 1 / α × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 + q 1 / α ,

using the fact that

Ω 3 ( q ; ξ ) ≔ ∫ 0 1 ξ ( 1 − q ξ ) β d q 0 ξ = ( 1 − q ) ∑ n = 0 ∞ q 2 n ( 1 − q n + 1 ) β .

This completes the proof of Theorem 4.5.□

Corollary 4.6

If β > 1 is taken to be positive integer, then under the assumption of Theorem 4.5, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( 2 , β + 1 ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 + q 1 / α .

Remark 4.3

If in Theorem 4.1 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1

G ( η 1 ) + G ( η 2 ) 2 − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d z ≤ ( η 2 − η 1 ) 2 2 . 2 1 / α B ( 2 , β + 1 ) 1 / β sup G ″ ( η 1 ) α , G ″ ( η 2 ) α 1 / α .

This coincides with the result in ref. [45].

Theorem 4.7

(4.7) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 [ q + 1 ] 1 / β × Ω 4 ( q ; ξ ) sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

where

Ω 4 ( q ; ξ ) ≔ ( 1 − q ) ∑ n = 0 ∞ q n ( 1 − q n + 1 ) α

and [β + 1] is the q-analogue of β + 1.

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

In view of Definition 2.11, we obtain

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 [ β + 1 ] 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α ∫ 0 1 ( 1 − q ξ ) α d q 0 ξ 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 [ β + 1 ] 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

using the fact that

Ω 4 ( q ; ξ ) ≔ ∫ 0 1 ( 1 − q ξ ) α d q 0 ξ = ( 1 − q ) ∑ n = 0 ∞ q n ( 1 − q n + 1 ) α .

This completes the proof of Theorem 4.7.□

Corollary 4.8

If α > 1 is taken to be positive integer, then under the assumption of Theorem 4.5, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 [ β + 1 ] 1 / β × B q ( α + 1 , 1 ) sup D q 2 η 1 G ( η 1 ) α + D q 2 η 1 G ( η 2 ) α 1 / α .

Remark 4.4

If in Theorem 4.1 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , we have a new result

(4.8) G ( η 1 ) + G ( η 2 ) 2 − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d z ≤ ( η 2 − η 1 ) 2 2 1 β + 1 1 / β × sup G ″ ( η 1 ) α + G ″ ( η 2 ) α α + 1 1 / α .

Theorem 4.9

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0), …, ϑ(κ), …) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-quasi-convex function on ϒ for α, β > 1, and α ⁻¹ + β ⁻¹ = 1, then

(4.9) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 5 ( q ; ξ ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α [ α + 1 ] 1 / α ,

where

Ω 5 ( q ; ξ ) ≔ ( 1 − q ) ∑ n = 0 ∞ q n ( 1 − q n + 1 ) β

and [α + 1] is the q-analogue of α + 1.

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

In view of Definition (2.11), we obtain

(4.10) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q Ω 5 ( q ; ξ ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α [ α + 1 ] 1 / α ,

using the fact that

Ω 5 ( q ; ξ ) ≔ ∫ 0 1 ( 1 − q ξ ) β d q 0 ξ = ( 1 − q ) ∑ n = 0 ∞ q n ( 1 − q n + 1 ) β .

This completes the proof of Theorem 4.7.□

Corollary 4.10

If β > 1 is taken to be a positive integer, then under the assumption of Theorem 4.9, we have

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , 1 ) 1 / β × sup D q 2 η 1 G ( η 1 ) α + D q 2 η 1 G ( η 2 ) α [ α + 1 ] 1 / α .

Remark 4.5

If in Theorem 4.9 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , then (4.9) reduces to (4.8).

Theorem 4.11

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0), …, ϑ(κ), …) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-quasi-convex function on ϒ for α, β > 1, and α ⁻¹ + β ⁻¹ = 1, then

(4.11) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( α + 1 , 2 ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α q + 1 1 / α .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

In view of Lemma 2.4 and using the fact that ( 1 − q ξ ) = ( 1 − q ξ ) q 1 , we obtain

(4.12) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q ∫ 0 1 ξ β ( 1 − q ξ ) q 1 d q 0 ξ 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 + q 1 / α = q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q B q ( β + 1 , 2 ) 1 / β × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 + q 1 / α .

This completes the proof of Theorem 4.11.□

Remark 4.6

If in Theorem 4.11 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , then (4.11) coincides with the result in ref. [45].

Theorem 4.12

Let s ∈ (0, 1], υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice q-differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that D q 2 η 1 G continuous on ϒ. If D q 2 η 1 G α is a generalized ψ-quasi-convex function on ϒ for α, β > 1 and α ⁻¹ + β ⁻¹ = 1, then

(4.13) q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 ( 1 + q ) 2 ( 1 + q + q 2 ) × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α .

Proof

Utilizing the fact that ∣ D q 2 η 1 G ∣ α is a generalized ψ-quasi-convex function with Hölder’s inequality and from Lemma 2.6, we have

In view of Lemma 2.5, we obtain

q G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 1 + q − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d q η 1 z ≤ q 2 ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 1 + q 1 ( 1 + q ) ( 1 + q + q 2 ) 1 − 1 / α × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α ( 1 + q ) ( 1 + q + q 2 ) 1 / α .

This completes the proof of Theorem 4.12.□

Remark 4.7

If in Theorem 4.12 it is taken limit when q → 1 and ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , then (4.13) coincides with the result in ref. [1].

5 Application

Specifically, in Definition 2.5 for ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , the generalized ψ-quasi-convex functions coincide with the quasi-convex functions. Moreover, if we put s = 1 along with ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) = η 2 − η 1 , in Definition 2.4, then the generalized ψ-convex functions reduce to the classical convex functions. Moreover, the q-integral inequalities would lead to the corresponding classical integral variants by selecting q ↦ 1⁻. Thus, various novel and earlier consequences can be obtained from Results in Sections 3 and 4 as special cases. Here, we illustrate the applications of our main results by further investigations.

Proposition 5.1

In the recent research, Zhuang et al. [46] established the q-integral inequalities for quasi-convex functions, the following inequality is stated as:

q G ( η 1 ) + G ( η 2 ) 1 + q − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d q η 1 z ≤ q 2 ( η 2 − η 1 ) 2 1 + q 1 1 + q 1 − 1 / α × Ω 1 ( q ; ξ ) sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α ,

where Ω₁(q; ξ) is defined as in (4.2).

In the following, we present a new analogous to inequality (4.1), which can be obtained directly by choosing q ↦ 1⁻ in Theorem 4.1.

Corollary 5.1

Let υ ₁, υ ₂ > 0 with a bounded sequence of real numbers ϑ = (ϑ(0),…, ϑ(κ),…) and 0 < q < 1. Suppose that a twice differentiable function G : ϒ ⊂ R → R defined on ϒ^∘ such that G ″ ∈ L 1 [ η 1 , η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ] with ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) > 0 . If G ″ α is a generalized ψ-quasi-convex function on ϒ for α ≥ 1 and α ⁻¹ + β ⁻¹ = 1. Then the following inequality holds: In the following, we present another new analogous to inequality (4.1), which can be obtained directly by choosing q ↦ 1⁻ in Theorem 4.1.

G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d z ≤ ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 4 2 ( α + 1 ) ( α + 2 ) 1 / α × sup G ″ ( η 1 ) α , G ″ ( η 2 ) α 1 / α .

Proposition 5.2

In ref. [46], Zhuang et al. derived another q-integral inequality for quasi-convex functions, the following inequality is stated as:

q G ( η 1 ) + G ( η 2 ) 1 + q − 1 η 2 − η 1 ∫ η 1 η 2 G ( z ) d q η 1 z ≤ q 2 ( η 2 − η 1 ) 2 ( 1 + q ) 2 ( 1 + q + q 2 ) × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 1 / α .

Corollary 5.2

G ( η 1 ) + G ( η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 − 1 ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ∫ η 1 η 1 + ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) G ( z ) d z ≤ ( ℱ υ 1 , υ 2 ϑ ( η 2 − η 1 ) ) 2 12 sup G ″ ( η 1 ) α , G ″ ( η 2 ) α 1 / α .

6 Conclusion

Recently, trapezoid-type inequalities have a significant contribution to the improvements of all areas of mathematical sciences. It has momentous investigations in the variability of applied analysis, for example, coding theory, geometric function theory, fractional calculus, impulsive diffusion equations and numerical analysis. Recently, several researchers have explored new ℋℋ -type variants for diverse kinds of convexities, preinvexities, statistical theory and so on. In the present investigation, we utilized a novel q-integral identity obtained in ref. [44] (Lemma 6) to explore certain quantum bounds for ℋℋ -type variants via newly introduced notions, generalized ψ-s-convex and generalized ψ-quasi-convex functions are elaborated. New theorem and new cases have been discussed in connection with Hölder and power mean inequality. Our consequences deduced several existing results in the related literature. Since q-calculus theory has potential utilities in special relativity theory, quantum mechanics and q-Hahn fractional calculus, we expect that this novel approach opens many avenues for interested researchers will endure discovering further quantum approximations of ℋℋ -type variants for other classes of convex functions, and, additionally, to discover uses in the aforementioned scientific disciplines.

Acknowledgements

The authors would like to express their sincere thanks to the referee and Editor.

Availability of supporting data: Not applicable. Competing interests: The authors declare that they have no competing interests.
Disclosure: The authors declare that they have no competing interests.
Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 61673169).
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Received: 2020-11-05

Revised: 2020-12-08

Accepted: 2021-01-14

Published Online: 2021-02-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems

Abstract

1 Introduction

2 Preliminaries

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4

Definition 2.5

Definition 2.6

Definition 2.7

Definition 2.8

Theorem 2.1

Definition 2.9

Definition 2.10

Definition 2.11

Lemma 2.2

Lemma 2.3

Lemma 2.4

Lemma 2.5

Lemma 2.6

3 Differentiable ℋℋ -type inequalities for generalized ψ-s-convex functions

Theorem 3.1

Proof

Corollary 3.2

Remark 3.1

Theorem 3.3

Proof

Corollary 3.4

Remark 3.2

Theorem 3.5

Proof

4 ℋℋ -type inequalities for generalized ψ-quasi-convex functions

Theorem 4.1

Proof

Corollary 4.2

Remark 4.1

Theorem 4.3

Proof

Corollary 4.4

Remark 4.2

Theorem 4.5

Proof

Corollary 4.6

Remark 4.3

Theorem 4.7

Proof

Corollary 4.8

Remark 4.4

Theorem 4.9

Proof

Corollary 4.10

Remark 4.5

Theorem 4.11

Proof

Remark 4.6

Theorem 4.12

Proof

Remark 4.7

5 Application

Proposition 5.1

Corollary 5.1

Proposition 5.2

Corollary 5.2

6 Conclusion

Acknowledgements

References

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