Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

Huo Tang; Kaliappan Vijaya; Gangadharan Murugusundaramoorthy; Srikandan Sivasubramanian

doi:10.1515/math-2021-0028

Open Access Published by De Gruyter Open Access May 20, 2021

Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

Huo Tang , Kaliappan Vijaya , Gangadharan Murugusundaramoorthy and Srikandan Sivasubramanian

From the journal Open Mathematics

https://doi.org/10.1515/math-2021-0028

Abstract

Let f k ( z ) = z + ∑ n = 2 k a n z n be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } , Re { f k ( z ) / f ( z ) } , Re { f ′ ( z ) / f k ′ ( z ) } and Re { f k ′ ( z ) / f ′ ( z ) } , where f ( z ) belongs to the subclass J p , q m ( μ , α , β ) of analytic functions, defined by Sălăgean ( p , q ) -differential operator. In addition, the inclusion relations involving N δ ( e ) of this generalized function class are considered.

Keywords: analytic; univalent; (p, q)-differential operator; partial sum; inclusion relation

MSC 2010: 30C45; 30C50

1 Introduction and preliminaries

Let A denote the class of functions of the form

(1.1) f ( z ) = z + ∑ n = 2 ∞ a n z n ,

which are analytic and univalent in the open disc D = { z : ∣ z ∣ < 1 } . We also denote T a subclass of A introduced and studied by Silverman [1], consisting of functions of the form:

(1.2) f ( z ) = z − ∑ n = 2 ∞ a n z n , a n ≥ 0 ; z ∈ D .

For functions f ∈ A given by (1.1) and g ∈ A given by g ( z ) = z + ∑ n = 2 ∞ b n z n , we define the Hadamard product (or convolution) of f and g by ( f ∗ g ) ( z ) = z + ∑ n = 2 ∞ a n b n z n , z ∈ D .

We briefly recall here the notion of q-operators i.e. q-difference operator that plays vital role in the theory of hypergeometric series, quantum physics and in the operator theory. The application of q-calculus was initiated by Jackson [2] (also see [3–5]). Kanas and Răducanu [4] have used the fractional q-calculus operators in investigations of certain classes of functions which are analytic in D . For p > 0 , q > 0 the (p; q)-differential operator of a function f , analytic in D is defined for a function f ∈ A is, by definition, given as follows [2]:

(1.3) D p , q f ( z ) = f ( p z ) − f ( q z ) ( p − q ) z ( p ≠ q ) .

From (1.3), we have

(1.4) D p , q f ( z ) = 1 + ∑ n = 2 ∞ [ n ] p , q a n z n − 1 ,

where

(1.5) [ n ] p , q = p n − q n p − q , [ 0 ] p , q = 0 .

The twin-basic number is a natural generalization of q -number, that is

(1.6) [ n ] q = 1 − q n 1 − q , ( q ≠ 1 ) ,

which is sometimes called the basic number n .

One can easily verify that D p , q f ( z ) → f ′ ( z ) as p → 1 − and q → 1 − . It is clear that q -integers and ( p ; q ) -integers differ, that is, we cannot obtain ( p ; q ) -integers just by replacing q by q p in the definition of q -integers. However, (1.5) reduces to (1.6) for the case p = 1 . Thus, we can say that ( p ; q ) -calculus can be taken as a generalization of q -calculus. The ( p ; q ) -factorial is defined by

(1.7) [ n ] p , q ! = ∏ k = 1 n [ k ] p , q ! ( n ≥ 1 ) , [ 0 ] p , q ! = 1 .

Note that p → 1 − the ( p ; q ) -factorial reduces to the q -factorial. Also, clearly

lim p → 1 − lim q → 1 − = [ n ] p , q ! = n ! and lim p → 1 − lim a → 1 − = [ n ] p , q = n .

For details on q-calculus and (p, q)-calculus, one can refer to [2,6,7] and also references cited therein. Recently for f ∈ A , Govindaraj and Sivasubramanian [8] defined Sălăgean (q)-differential operator and further Kanas and Răducanu [4] defined and discussed Sălăgean (p, q)-differential operator as follows:

(1.8) D p , q 0 f ( z ) = f ( z ) , D p , q 1 f ( z ) = z D p , q f ( z ) , D p , q m f ( z ) = z D p , q m ( D p , q m − 1 f ( z ) ) , D p , q m f ( z ) = z + ∑ n = 2 ∞ [ n ] p , q m a n z n ( m ∈ N 0 , z ∈ D ) .

It is interesting that one can observe

(1.9) lim ( p , q ) → ( 1 − , 1 − ) D p , q m f ( z ) = D m f ( z ) = z + ∑ n = 2 ∞ n m a n z n ( m ∈ N 0 , z ∈ D ) ,

the familiar Sălăgean derivative [9].

For 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 , we let J p , q m ( μ , α , β ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.10) ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − α > β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 , z ∈ D ,

where D p , q m f ( z ) is given by (1.8). We further let TJ p , q m ( μ , α , β ) = J p , q m ( μ , α , β ) ∩ T .

By taking μ = 1 we get TJ p , q m ( 1 , α , β ) ≡ TSP p , q m ( α , β ) studied by Kanas and Răducanu [4]. Further by specializing the parameter μ = 0 , we define the following new subclass:

Remark 1.1

For μ = 0 , m ∈ N 0 , 0 ≤ α < 1 and β ≥ 0 , let TJ p , q m ( 0 , α , β ) ≡ USD p , q m ( α , β ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.11) ℜ D p , q m + 1 f ( z ) z − α > β D p , q m + 1 f ( z ) z − 1 , z ∈ D ,

where D p , q m f ( z ) is given by (1.8).

Remark 1.2

For μ = 0 , β = 0 , 0 ≤ α < 1 and m ∈ N 0 , let TJ p , q m ( 1 , α , 0 ) ≡ ℛ p , q m ( α ) be the subclass of A , consisting of functions of the form (1.1) and satisfying the analytic criterion

(1.12) ℜ D p , q m + 1 f ( z ) z > α z ∈ D ,

where D p , q m f ( z ) is given by (1.8).

Remark 1.3

As lim ( p , q ) → ( 1 − , 1 − ) , D p , q m f ( z ) reduces to various interesting subclasses (as mentioned earlier) satisfying the analytic criterion

(1.13) ℜ D m + 1 f ( z ) ( 1 − μ ) z + μ D m f ( z ) − α > β D m f ( z ) ( 1 − μ ) z + μ D m f ( z ) − 1 , z ∈ D ,

where D m f ( z ) is given by (1.9).

2 Basic properties

In this section, we obtain the characterization properties for the classes TJ p , q m ( μ , α , β ) .

Theorem 2.1

A function f ( z ) of the form (1.1) is in J p , q m ( μ , α , β ) if

(2.1) ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) ∣ a n ∣ ≤ 1 − α ,

where 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 .

The result is sharp for the function

f n ( z ) = z − 1 − α [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) z n .

Proof

It suffices to show that

β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 − ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ 1 − α .

We have

β D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 − ℜ D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ ( 1 + β ) D p , q m + 1 f ( z ) ( 1 − μ ) z + μ D p , q m f ( z ) − 1 ≤ ( 1 + β ) ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q − μ ) ∣ a n ∣ ∣ z ∣ n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ ∣ a n ∣ ∣ z ∣ n − 1 .

As ∣ z ∣ → 1 − , the last expression is bounded above by 1 − α if (2.1) holds. It is obvious that the function f n satisfies the inequality (2.1), and thus 1 − α cannot be replaced by a larger number. Therefore, we need only to prove that f ∈ TJ p , q m ( μ , α , β ) . Since

ℜ 1 − ∑ n = 2 ∞ [ n ] p , q m + 1 a n z n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ a n z n − 1 − α > β ∑ n = 2 ∞ [ n ] p , q m ( [ n ] p , q − μ ) a n z n − 1 1 − ∑ n = 2 ∞ [ n ] p , q m μ a n z n − 1 .

Letting z → 1 along the real axis, we obtain the desired inequality given in (2.1).□

Corollary 2.2

If f ∈ TJ p , q m ( μ , α , β ) , then

(2.2) ∣ a n ∣ ≤ 1 − α Φ p , q m ( μ , α , β ) .

Equality holds for the function f ( z ) = z − 1 − α Φ p , q m ( μ , α , β ) z n , where

Φ p , q m ( μ , α , β ) = [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) ) .

Throughout this paper for convenience, unless otherwise stated, we let

(2.3) Φ n m = Φ p , q m ( μ , α , β ) = [ n ] p , q m ( [ n ] p , q ( 1 + β ) − μ ( α + β ) )

and

(2.4) Φ 2 m = Φ p , q m ( μ , α , β ) = [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ,

where 0 ≤ μ ≤ 1 , 0 ≤ α < 1 , β ≥ 0 and m ∈ N 0 .

3 Partial sums

Silverman [10] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. In this section, following the earlier work by Silverman [10] and also the work cited in [11–15] on partial sums of analytic functions, we study the ratio of a function of the form (1.1) to its sequence of partial sums of the form

f k ( z ) = z + ∑ n = 2 k a n z n ,

when the coefficients of f ( z ) satisfy the condition (2.1).

Theorem 3.1

If f ∈ A of the form (1.1) satisfies the condition (2.1), then

(3.1) ℜ f ( z ) f k ( z ) ≥ Φ k + 1 m − 1 + α Φ k + 1 m ( z ∈ D ) ,

where

(3.2) Φ n m = Φ p , q m ( μ , α , β ) ≥ 1 − α , i f n = 2 , 3 , … , k , Φ k + 1 m , i f n = k + 1 , k + 2 , … .

The result (3.1) is sharp with the function given by

(3.3) f ( z ) = z + 1 − α Φ k + 1 m z k + 1 .

Proof

Define the function w ( z ) by

(3.4) 1 + w ( z ) 1 − w ( z ) = Φ k + 1 m 1 − α f ( z ) f k ( z ) − Φ k + 1 m − 1 + α Φ k + 1 m = 1 + ∑ n = 2 k a n z n − 1 + Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 1 + ∑ n = 2 k a n z n − 1 .

It suffices to show that ∣ w ( z ) ∣ ≤ 1 . Now, from (3.4) we can write

w ( z ) = Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 2 + 2 ∑ n = 2 k a n z n − 1 + Φ k + 1 m 1 − α ∑ n = k + 1 ∞ a n z n − 1 .

Hence, we obtain

∣ w ( z ) ∣ ≤ Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ 2 − 2 ∑ n = 2 k ∣ a n ∣ − Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ .

Now ∣ w ( z ) ∣ ≤ 1 if and only if

2 Φ k + 1 m 1 − α ∑ n = k + 1 ∞ ∣ a n ∣ ≤ 2 − 2 ∑ n = 2 k ∣ a n ∣ ,

or, equivalently,

∑ n = 2 k ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k + 1 m 1 − α ∣ a n ∣ ≤ 1 .

From the condition (2.1), it is sufficient to show that

∑ n = 2 k ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k + 1 m 1 − α ∣ a n ∣ ≤ ∑ n = 2 ∞ Φ n m 1 − α ∣ a n ∣ ,

which is equivalent to

(3.5) ∑ n = 2 k Φ k m − 1 + α 1 − α ∣ a n ∣ + ∑ n = k + 1 ∞ Φ k m − Φ k + 1 m 1 − α ∣ a n ∣ ≥ 0 .

To see that the function given by (3.3) gives the sharp result, we observe that for z = r e i π / n

f ( z ) f k ( z ) = 1 + 1 − α Φ k + 1 m z n → 1 − 1 − α Φ k + 1 m = Φ k + 1 m − 1 + α Φ k + 1 m , when r → 1 − . □

Theorem 3.2

If f of the form (1.1) satisfies the condition (2.1), then

(3.6) ℜ f k ( z ) f ( z ) ≥ Φ k + 1 m Φ k + 1 m + 1 − α ( z ∈ D ) ,

where Φ n + 1 m ≥ 1 − α and

(3.7) Φ k m ≥ 1 − α , i f n = 2 , 3 , … , k , Φ k + 1 m , i f n = k + 1 , k + 2 , … .

The result (3.6) is sharp with the function given by (3.3).

Proof

The proof follows by defining

1 + w ( z ) 1 − w ( z ) = Φ k + 1 m + 1 − α 1 − α f k ( z ) f ( z ) − Φ k + 1 m Φ k + 1 m + 1 − α

and much akin to similar arguments in Theorem 3.1.□

We next turn to ratios involving derivatives.

Theorem 3.3

If f of the form (1.1) satisfies the condition (2.1), then

(3.8) ℜ f ′ ( z ) f n ′ ( z ) ≥ Φ n + 1 m − ( n + 1 ) ( 1 − α ) Φ n + 1 m ( z ∈ D )

and

(3.9) ℜ f n ′ ( z ) f ′ ( z ) ≥ Φ n + 1 m Φ n + 1 m + ( n + 1 ) ( 1 − γ ) ( z ∈ D ) ,

where Φ n + 1 m ≥ ( n + 1 ) ( 1 − α ) and

(3.10) Φ k m ≥ k ( 1 − α ) , i f k = 2 , 3 , … , n , k Φ n + 1 m n + 1 , i f k = n + 1 , n + 2 , … .

The results are sharp with the function given by (3.3).

Proof

We write

1 + w ( z ) 1 − w ( z ) = Φ n + 1 m ( n + 1 ) ( 1 − α ) f ′ ( z ) f n ′ ( z ) − Φ n + 1 m − ( n + 1 ) ( 1 − α ) Φ n + 1 m ,

where

w ( z ) = Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k a k z k − 1 2 + 2 ∑ k = 2 n k a k z k − 1 + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k a k z k − 1 .

Now ∣ w ( z ) ∣ ≤ 1 if and only if

∑ k = 2 n k ∣ a k ∣ + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k ∣ a k ∣ ≤ 1 .

From the condition (2.1), it is sufficient to show that

∑ k = 2 n k ∣ a k ∣ + Φ n + 1 m ( n + 1 ) ( 1 − α ) ∑ k = n + 1 ∞ k ∣ a k ∣ ≤ ∑ k = 2 ∞ Φ k m 1 − γ ∣ a k ∣ ,

which is equivalent to

∑ k = 2 n Φ k m − ( 1 − α ) k 1 − α ∣ a k ∣ + ∑ k = n + 1 ∞ ( n + 1 ) Φ k m − k Φ n + 1 m ( n + 1 ) ( 1 − α ) ∣ a k ∣ ≥ 0 .

To prove the result (3.9), we define the function w ( z ) by

1 + w ( z ) 1 − w ( z ) = ( n + 1 ) ( 1 − α ) + Φ n + 1 m ( 1 − α ) ( n + 1 ) f n ′ ( z ) f ′ ( z ) − Φ n + 1 m ( n + 1 ) ( 1 − α ) + Φ n + 1 m

and by similar arguments in the first part we get desired result.□

4 Inclusion relations involving N δ ( e )

In this section following [16–18], we define the n , δ neighborhood of function f ( z ) ∈ T and discuss the inclusion relations involving N δ ( e ) .

(4.1) N δ ( f ) = g ∈ T : g ( z ) = z − ∑ n = 2 ∞ b n z n and ∑ n = 2 ∞ n ∣ a n − b n ∣ ≤ δ .

Particularly for the identity function e ( z ) = z , we have

(4.2) N δ ( e ) = g ∈ T : g ( z ) = z − ∑ n = 2 ∞ b n z n and ∑ n = 2 ∞ n ∣ b n ∣ ≤ δ .

Theorem 4.1

Let

(4.3) δ = 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

Then TJ p , q m ( μ , α , β ) ⊂ N δ ( e ) .

Proof

For f ∈ TJ p , q m ( μ , α , β ) , Theorem 2.1, yields

[ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ∑ n = 2 ∞ a n ≤ 1 − α ,

so that

(4.4) ∑ n = 2 ∞ a n ≤ 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

On the other hand, from (2.1) and (4.4) we have

(4.5) [ 2 ] p , q m ( 1 + β ) ∑ n = 2 ∞ [ n ] p , q a n ≤ 1 − α + [ 2 ] p , q m μ ( α + β ) ∑ n = 2 ∞ a n ≤ 1 − α + [ 2 ] p , q m μ ( α + β ) ( 1 − α ) [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) ≤ [ 2 ] p , q m ( 1 + β ) ( 1 − α ) [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ∑ n = 2 ∞ [ n ] p , q a n ≤ 1 − α [ 2 ] p , q ( 1 + β ) − μ ( α + β ) .

□

Now we determine the neighborhood for each of the class TJ p , q m ( μ , α , β ) , which we define as follows.

A function f ∈ T is said to be in the class TJ p , q m ( μ , α , β ) if there exists a function g ∈ TJ m ( μ , α , β , η ) such that

(4.6) f ( z ) g ( z ) − 1 < 1 − η , ( z ∈ D , 0 ≤ η < 1 ) .

Theorem 4.2

If g ∈ TJ p , q m ( μ , α , β ) and

(4.7) η = 1 − δ [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) 2 [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

Then N δ ( g ) ⊂ TJ p , q m ( μ , α , β , η ) .

Proof

Suppose that f ∈ N δ ( g ) , then we find from 4.1 that

∑ n = 2 ∞ n ∣ a n − b n ∣ ≤ δ ,

which implies that the coefficient inequality

∑ n = 2 ∞ ∣ a n − b n ∣ ≤ δ 2 .

Next, since g ∈ TJ p , q m ( μ , α , β ) , we have

∑ n = 2 ∞ b n ≤ 1 − α [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) .

So that

f ( z ) g ( z ) − 1 < ∑ n = 2 ∞ ∣ a n − b n ∣ 1 − ∑ n = 2 ∞ b n ≤ δ 2 × [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) [ 2 ] p , q m ( [ 2 ] p , q ( 1 + β ) − μ ( α + β ) ) − ( 1 − α ) ≤ 1 − η ,

provided that η is given precisely by (4.7). Thus by definition, f ∈ TJ p , q m ( μ , α , β , η ) for η given by (4.7), which completes the proof.□

5 Concluding remarks and observations

As a special case of the aforementioned theorems, we can determine new sharp lower bounds for ℜ f ( z ) f n ( z ) , ℜ f n ( z ) f ( z ) , ℜ f ′ ( z ) f n ′ ( z ) and ℜ f n ′ ( z ) f ′ ( z ) for various function classes stated in Remarks 1.1 and 1.2 and upon specializing the values of μ and β one can deduce various new subclasses on p , q -difference operator and prove the above partial sums and neighborhood results.

Funding information: The research of Huo Tang was partly supported by the Natural Science Foundation of the People’s Republic of China under Grants 11561001 and 11271045, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People’s Republic of China under Grant 2018MS01026 and the Natural Science Foundation of Chifeng of Inner Mongolia. The work of S. Sivasubramanian was supported by a grant from the Science and Engineering Research Board, Government of India under Mathematical Research Impact Centric Support of Department of Science and Technology (DST) (vide ref: MTR/2017/000607).
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflict of interest: Authors state no conflict of interest.

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Received: 2020-07-09

Revised: 2021-01-07

Accepted: 2021-01-07

Published Online: 2021-05-20

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

Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

Abstract

1 Introduction and preliminaries

Remark 1.1

Remark 1.2

Remark 1.3

2 Basic properties

Theorem 2.1

Proof

Corollary 2.2

3 Partial sums

Theorem 3.1

Proof

Theorem 3.2

Proof

Theorem 3.3

Proof

4 Inclusion relations involving N δ ( e )

Theorem 4.1

Proof

Theorem 4.2

Proof

5 Concluding remarks and observations

References

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