Introduction and preliminary results

Any function f be continuous on an interval I such that its value at the midpoint of the interval does not exceed from the arithmetic mean of boundary values of the interval, termed as a convex function. The investigation of an important mathematical problem informs how function behaves using means. Jensen convex function is one of the eminent cases that deals with arithmetic mean [1, pp.2]. A function f is convex on an interval [p, q] if

$$f(\alpha ~\eta +\beta ~\zeta )\le \alpha f(\eta )+\beta f(\zeta ),$$

where \(\eta ,\zeta \in [p,q]\) and \(\alpha ,\beta \in ~]0,1]\). Further, a real-valued function defined on a non-empty subinterval I of \({\mathbb {R}}\) is called convex if we replace \(\alpha +\beta =1\) for all points \(\eta ,\zeta \in I\) in the above inequality. It is called strictly convex if the above inequality holds strictly whenever \(\eta \) and \(\zeta \) are distinct points. In addition, \(-f\) is convex/ strictly convex; accordingly, f is concave/ strictly concave. Further, a function is called affine if it is both convex and concave. The application of Hermite–Hadamard-type inequalities and convexities can be found in [2,3,4,5,6,7].

In the field of applied sciences, the occurrence of new mathematical inequalities puts the foundation for the heuristic algorithms. Hermite–Hadamard’s is one of the main inequalities that yields explicitly the error bounds of the trapezoidal and midpoint rules for a smooth convex function \(f:[p,q] \rightarrow {\mathbb {R}}\), defined as

$$ f\left( \frac{p+q}{2}\right) \le \frac{1}{q-p}\int _p^q f(\tau )d\tau \le \frac{f(p)+f(q)}{2}. $$
(1.1)

For f to be concave, these inequalities also hold in reverse order. More precisely, Hermite–Hadamard’s inequality (1.1) may depict the concept of convexity and follows from Jensen inequality. Inequality (1.1) has gained much attention among researchers due to its remarkable characteristics in refinements and generalizations, as well. Simpson’s is another well-known type of inequality, defined as

$$\begin{aligned}&\left| \frac{1}{3}\left[ 2f\left( \frac{p+q}{2}\right) +\frac{f(p)+f(q)}{2}\right] \right. \\&\left. \quad -\,\frac{1}{q-p}\int _p^q f(\eta )d\eta \right| \le \frac{1}{2880}\parallel f^{(iv)}\parallel _\infty (q-p)^4, \end{aligned}$$
(1.2)

where \(f:[p,q] \rightarrow {\mathbb {R}}\) is a four times continuous differentiable mapping on (p, q) and \(\parallel f^{(iv)}\parallel _\infty = \sup _{\eta \in (p, q)}\left| f^{(iv)}(\eta )\right| <\infty \). It is to be mentioned that the classical Simpson quadrature formula cannot be applied for f is neither differentiable four times nor \(f^{(iv)}\) bounded on (p, q).

Moreover, the quasi-convex function, \(f:[p,q]\rightarrow {\mathbb {R}}\), is the generalization of convex function, defined as

$$ f(\alpha \eta +\beta \zeta )\le \sup \{f(\eta ),f(\zeta )\}, $$

where \(\eta ,\zeta \in [p,q]\) and \(\alpha ,\beta \in ]0,1]\) such that \(\alpha +\beta =1\).

Initially, the notion of geometrically convex was introduced by Niculescu in [8, 9] and produced as

Definition 1

A function \(f:I\subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+\) is said to be GG-convex (called geometrically convex function) if

$$ f\left( \eta ^\alpha \zeta ^\beta \right) \le f^{\alpha }(\eta )f^{\beta }(\zeta ),$$

where \(\eta ,\zeta \in [p,q]\) and \(\alpha ,\beta \in ]0,1]\) such that \(\alpha +\beta =1\).

Niculescu, in the same article, defined the term \(geometric-arithmatically\) convex with notation GA-convex, as

Definition 2

A function \(f:I\subseteq {\mathbb {R}}^+\rightarrow {\mathbb {R}}\) is termed as GA-convex (called geometrically convex function) if

$$ f\left( \eta ^\alpha \zeta ^\beta \right) \le \alpha f(\eta )+\beta f(\zeta ), $$

where \(\eta ,\zeta \in [p,q]\) and \(\alpha ,\beta \in ]0,1]\) such that \(\alpha +\beta =1\).

The following definition of quasi-geometrically convex functions was first reported by İşcan [10]

Definition 3

A function \(f:I\subseteq {\mathbb {R}}^+\rightarrow {\mathbb {R}}\) is said to be quasi-convex if

$$ f\left( \eta ^\alpha \zeta ^\beta \right) \le \sup \{f(\eta ),f(\zeta )\},$$

where \(\eta ,\zeta \in [p,q]\) and \(\alpha ,\beta \in ]0,1]\) such that \(\alpha +\beta =1\).

Both GA-convex and geometrically convex functions are quasi-geometrically convex functions, but there exist quasi-geometrically convex functions which are neither GA-convex nor GG-convex discussed in [11, 12].

Recently, İşcan et al. [13] developed some results based on single differentiability for quasi-geometrically convex functions using the identity

Lemma 1

A function\(f:I\subseteq {\mathbb {R}}^+\rightarrow {\mathbb {R}}\)be a differentiable function on\(I^o\)such that\(f\in L^1([p,q])\), where\(p,q\in I\)with\(p<q\). Then for all\(\lambda ,\mu \in {\mathbb {R}}\), we have

$$\begin{aligned}&I_f(\lambda ,\mu ,p,q)=ln\left( \frac{q}{p}\right) \\&\quad \times \,\left\{ \int _0^{\frac{1}{2}}(\tau -\mu )p^{1-\tau }q^\tau f'\left( p^{1-\tau }q^\tau \right) d\tau \right. \\&\quad \left. +\,\int _{\frac{1}{2}}^1(\tau -\lambda )p^{1-\tau }q^\tau f'\left( p^{1-\tau }q^\tau \right) d\tau \right\} , \end{aligned}$$
(1.3)

where

$$\begin{aligned}&I_f\left( \lambda ,\mu ,p,q\right) =\left( \lambda -\mu \right) f\left( \sqrt{pq}\right) +\mu f(p) + \left( 1-\lambda \right) f(q)\\&\quad -\,\frac{1}{ln\left( \frac{q}{p}\right) }\int _p^q \frac{f(\xi )}{\xi }d\xi . \end{aligned}$$

This paper is in the continuation of [13]. The main purpose of the paper is to develop some new integral inequalities of both Hadamard and Simpson type for twice differentiable mappings using new integral identity.

Main results

The following identity is needed to prove main results.

Lemma 2

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a differentiable function on\(I^o\)such that\(f \in L^1([p,q])\),where\(p,q\in I\)with\(p<q\). Then for all\(\lambda ,\mu \in {\mathbb {R}}\), we have

$$\begin{aligned}&|M_f(\lambda , \mu ,p,q)|=\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times \,\left\{ \int _0^{\frac{1}{2}}\tau (\tau -\mu )p^{2(1-\tau )}q^{2\tau }f''\left( p^{1-\tau }q^\tau \right) d\tau \right. \\&\quad \left. +\,\int _{\frac{1}{2}}^1(1-\tau )(\tau -\lambda )p^{2(1-\tau )}q^{2\tau }f''\left( p^{1-\tau }q^\tau \right) d\tau \right\} , \end{aligned}$$
(2.1)

where

$$\begin{aligned}&M_f\left( \lambda ,\mu ,p,q\right) =\left( \lambda -\mu +1\right) f\left( \sqrt{pq}\right) \\&\quad +\,\mu f(p)+\left( 1-\lambda \right) f(q)\\&\quad +\,\frac{\sqrt{pq}\left( \lambda +\mu -1\right) }{2}ln\left( \frac{q}{p}\right) f'\left( \sqrt{pq}\right) \\&\quad -\,\frac{2}{ln\left( q/p\right) }\int _p^q \frac{f(\xi )}{\xi }d\xi . \end{aligned}$$

Proof

Using integration rules and changing parameter, it leads to the result. \(\square \)

Theorem 1

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\) such that \(f''\in L^1([p,q])\), where\(p,q\in I\)with\(p<q\). If \(|f''|^k\) is quasi-geometrically convex on [p, q] for some fixed\(k \ge 1\) and \(0 \le \mu \le 1/ 2 \le \lambda \le 1\), then the following inequality holds

$$\begin{aligned}&\left| M_f\left( \lambda ,\mu ,p,q\right) \right| \le \left( \ln \left( \frac{q}{p}\right) \right) \\&\quad \times\, \left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}}\left\{ c_1^{1-\frac{1}{k}}(\mu )c_3^{\frac{1}{k}}(\mu ,k,p,q)\right. \\&\quad \left. +\,c_2^{1-\frac{1}{k}}(\lambda )c_4^{\frac{1}{k}}(\lambda , k,p,q)\right\} , \end{aligned}$$
(2.2)

where

$$\begin{aligned}&c_1(\mu ) = \frac{\mu ^3}{3}-\frac{\mu }{8}+\frac{1}{24}, \\&\quad c_2(\lambda ) = -\frac{\lambda ^3}{3}+\lambda ^2-\frac{7\lambda }{8}+\frac{1}{4}, \\&\quad c_3(\mu ,k,p,q) = \frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times\, \left[ 8\mu ^2p^{2k(1-\mu )}\mathrm {L}\left( p^{2k\mu }, q^{2k\mu }\right) -\frac{8\mu p^{2k(1-\mu )}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. \times \,\mathrm {L}\left( p^{2k\mu }, q^{2k\mu }\right) +\frac{p^k}{k\left( \ln \left( \frac{q}{p}\right) \right) }\mathrm {L}\left( p^k, q^k\right) \right. \\&\left. \quad +\,\frac{10\mu p^{2k}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. +\,\frac{2(\mu -1)}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }(pq)^k+\frac{1-2\mu }{2}(pq)^k\right] , \end{aligned}$$

and

$$\begin{aligned}&c_4(\lambda ,k,p,q) = \frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times\, \left[ \frac{2(1-\lambda )q^{2k}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }-\frac{(2\lambda -1)(pq)^k}{2}+4\lambda (1-\lambda )p^{2k(1-\lambda )} \right. \\&\quad \left. \times\, \mathrm {L}\left( p^{2k\lambda }, q^{2k\lambda }\right) -2\lambda p^k\mathrm {L}\left( p^k, q^k\right) \right. \\&\quad \left. +\,\frac{8p^{2k(1-\lambda )}q^{2k\lambda }}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}+\frac{2\lambda (pq)^k}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. -\,\frac{4q^{2k}}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}-\frac{4(pq)^k}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}\right] . \end{aligned}$$

Proof

Since \(|f''|^k\) is quasi-geometrically convex on [p, q] for all \(\tau \in [0,1]\)

$$\left| f''\left( p^{1-\tau }q^\tau \right) \right| ^k \le \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} $$

Using Lemma 2 and power mean inequality, it yields

$$\begin{aligned}&\left| M_f(\lambda , \mu ,p,q)\right| =\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times\, \left\{ \left( \int _0^{\frac{1}{2}}|\tau (\tau -\mu )|d\tau \right) ^{1-\frac{1}{k}}\left( \int _0^{\frac{1}{2}}|\tau (\tau -\mu )|\left( p^{2(1-\tau )}q^{2\tau }\right) ^k \right. \right. \\&\left. \left. \quad \times\, \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} \right) ^{\frac{1}{k}}\right. \\&\left. \quad +\,\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|d\tau \right) ^{1-\frac{1}{k}}\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|\right. \right. \\&\left. \left. \quad \times\, \left( p^{2(1-\tau )}q^{2\tau }\right) ^k\sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} \right) ^{\frac{1}{k}}\right\} . \end{aligned}$$
(2.3)

Here we assume that

$$\begin{aligned}&c_1(\mu ) = \int _0^{\frac{1}{2}}|\tau (\tau -\mu )|d\tau =\int _0^\mu (\tau (\mu -\tau ))d\tau \\&\qquad +\, \int _\mu ^{\frac{1}{2}}(\tau (\tau -\mu ))d\tau =\frac{\mu ^3}{3}-\frac{\mu }{8}+\frac{1}{24},\\&\quad c_2(\lambda ) = \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|d\tau =\int _{\frac{1}{2}}^\lambda (1-\tau )(\lambda -\tau )d\tau \\&\qquad + \,\int _\lambda ^1(1-\tau )(\tau -\lambda )d\tau =-\frac{\lambda ^3}{3}\\&\qquad +\,\lambda ^2-\frac{7\lambda }{8}+\frac{1}{4},\\&\quad c_3(\mu ,k,p,q) = \int _0^{\frac{1}{2}}|\tau (\mu -\tau )|\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau \\&\quad =\int _0^\mu \tau (\mu -\tau )\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau \\&\qquad + \,\int _\mu ^{\frac{1}{2}}\tau (\tau -\mu )\left( a^{2(1-\tau )}q^{2\tau }\right) ^kd\tau . \end{aligned}$$

Using substitution \(\xi =p^{2(1-\tau )}q^{2\tau }\) in \(c_3(\mu ,k,p,q)\), it leads to

$$\begin{aligned}&\int _0^\mu \tau (\mu -\tau )\left( p^{2(1-\tau )}q^{2\tau }\right) d\tau \\&\quad =\frac{\mu }{\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}\int _{p^2}^{p^{2(1-\mu )}q^{2\mu }}\xi ^{k-1}\ln \left( \frac{\xi }{p^2}\right) d\xi \\&\qquad -\,\frac{\mu }{\left( 2\ln \left( \frac{q}{p}\right) \right) ^3}\int _{p^2}^{p^{2(1-\mu )}q^{2\mu }}\xi ^{k-1}\ln \left( \frac{\xi }{p^2}\right) d\xi \\&\int _0^\mu \tau (\mu -\tau )\left( p^{2(1-\tau )}q^{2\tau }\right) d\tau \\&\quad =\frac{\mu p^{2k(1-\mu )}q^{2k\mu }}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}-\frac{2 p^{2k(1-\mu )}q^{2k\mu }}{k^3\left( 2\ln \left( \frac{q}{p}\right) \right) ^3}\\&\qquad +\,\frac{\mu p^{2k}}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}+\frac{\mu p^{2k}}{k^3\left( 2\ln \left( \frac{q}{p}\right) \right) ^3}\\&\int _\mu ^{\frac{1}{2}} \tau (\tau -\mu )\left( p^{2(1-t)}q^{2\tau }\right) d\tau \\&\quad =\frac{1}{\left( 2\ln \left( \frac{q}{p}\right) \right) ^3}\int _{p^{2(1-\mu )}q^{2\mu }}^{pq}\xi ^{k-1}\left( \ln \left( \frac{\xi }{p^2}\right) \right) ^2d\xi \\&\qquad -\,\frac{\mu }{\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}\int _{p^{2(1-\mu )}q^{2\mu }}^{pq}\xi ^{k-1}\ln \left( \frac{\xi }{p^2}\right) d\xi \\&\int _\mu ^{\frac{1}{2}} \tau (\tau -\mu )\left( p^{2(1-\tau )}q^{2\tau }\right) d\tau \\&\quad =\frac{(1-2\mu ) (pq)^k}{8k\ln \left( \frac{q}{p}\right) }+\frac{(\mu -1) (pq)^k}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}+\frac{3\mu p^{2k(1-\mu )}q^{2k\xi }}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}\\&\qquad +\,\frac{2 (pq)^k}{k^3\left( 2\ln \left( \frac{q}{p}\right) \right) ^3}-\frac{2 p^{2k(1-\mu )}q^{2k\mu }}{k^3\left( 2\ln \left( \frac{p}{q}\right) \right) ^3} . \end{aligned}$$

Finally, we get

$$\begin{aligned}&c_3(\mu ,k,p,q) = \frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times \,\left[ 8\mu ^2p^{2k(1-\mu )}\mathrm {L}\left( p^{2k\mu }, q^{2k\mu }\right) -\frac{8\mu p^{2k(1-\mu )}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. \times\, \mathrm {L}\left( p^{2k\mu }, q^{2k\mu }\right) +\frac{p^k}{k\left( \ln \left( \frac{q}{p}\right) \right) }\mathrm {L}\left( p^k, q^k\right) +\frac{10\mu p^{2k}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. +\,\frac{2(\mu -1)}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }(pq)^k+\frac{1-2\mu }{2}(pq)^k\right] . \end{aligned}$$

And

$$\begin{aligned} c_4(\lambda ,k,p,q)&= {} \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau \\ &= {} \int _{\frac{1}{2}}^\lambda \tau (\mu -\tau )\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau \\&+ \int _\lambda ^1\tau (\tau -\mu )\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau . \end{aligned}$$

Using same substitution \(\xi =p^{2(1-\tau )}q^{2\tau }\) in \(c_4(\lambda ,k,p,q)\), we have

$$\begin{aligned}&c_4(\lambda ,k,p,q) = \frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times \,\left[ \frac{2(1-\lambda )q^{2k}}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }-\frac{(2\lambda -1)(pq)^k}{2}+4\lambda (1-\lambda )p^{2k(1-\lambda )} \right. \\&\quad \left. \times\, \mathrm {L}\left( p^{2k\lambda }, q^{2k\lambda }\right) -2\lambda p^k\mathrm {L}\left( p^k, q^k\right) \right. \\&\left. \quad +\,\frac{8p^{2k(1-\lambda )}q^{2k\lambda }}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}+\frac{2\lambda (pq)^k}{k\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\quad \left. -\,\frac{4q^{2k}}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}-\frac{4(pq)^k}{k^2\left( 2\ln \left( \frac{q}{p}\right) \right) ^2}\right] . \end{aligned}$$

This completes the proof. \(\square \)

Corollary 1

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\)such that\(f''\in L^1([p,q])\), where\(p,q\in I\)with\(p<q\). If\(|f''|^k\)isquasi-geometrically convex on [p, q] for some fixed\(k \ge 1\)and for\(l, m \in {\mathbb {R}}\)with\(l<m\), then the following inequality holds

$$\begin{aligned}&\left| m_f\left( \frac{l}{m},p,q\right) \right| \le \frac{8l^3-3m^2l+m^3}{24m^3}\left( \ln \left( \frac{q}{p}\right) \right) \\&\quad \times\, \left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}} \\&\quad \times \,\left\{ c_3^{\frac{1}{q}}\left( \frac{l}{m},k,p, q\right) +c_4^{\frac{1}{k}}\left( \frac{l}{m},k,p,q\right) \right\} , \end{aligned}$$
(2.4)

where

$$\begin{aligned}&m_f\left( \frac{l}{m},p, q\right) =2\left( \frac{m-l}{m}\right) f\left( \sqrt{pq}\right) \\&\quad +\,\frac{l}{m}\left( f(p)+f(q)\right) -\frac{2}{ln\left( \frac{q}{p}\right) }\int _p^q \frac{f(\xi )}{\xi }d\xi , \end{aligned}$$

and

$$\begin{aligned}&c_1\left( \frac{l}{m}\right) = c_2\left( \frac{l}{m}\right) =\frac{8l^3-3m^2l+m^3}{24m^3},\\&c_3\left( \frac{l}{m},k,p,q\right) = \frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times \,\left[ \frac{8l}{m}p^{\frac{2k(m-l)}{m}}\left( \frac{lk\left( 2\ln \left( \frac{q}{p}\right) \right) -m}{mq\left( 2\ln \left( \frac{q}{p}\right) \right) }\right) \mathrm {L}\left( p^{2k\frac{l}{m}}, q^{2k\frac{l}{m}}\right) \right. \\&\quad \left. +\,\frac{p^k}{k\left( \ln \left( \frac{q}{p}\right) \right) }\mathrm {L}\left( p^k, q^k\right) +\frac{10lp^{2k}}{mq\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\left. \quad -\,\frac{2(m-l)}{mk\left( 2\ln \left( \frac{q}{p}\right) \right) }(pq)^k+\frac{(m-2l)}{2m}(pq)^k\right] , \\&c_4\left( \frac{l}{m},k,p,q\right) =\frac{1}{2k\left( 2\ln \left( \frac{q}{p}\right) \right) }\\&\quad \times\, \left[ \frac{8l}{m}b^{\frac{2k(m-l)}{m}}\left( \frac{lk\left( 2\ln \left( \frac{q}{p}\right) \right) -m}{mk\left( 2\ln \left( \frac{q}{p}\right) \right) }\right) \mathrm {L}\left( p^{2k\frac{l}{m}}, q^{2k\frac{l}{m}}\right) \right. \\&\quad \left. +\,\frac{q^k}{k\left( \ln \left( \frac{q}{p}\right) \right) }\mathrm {L}\left( p^k, q^k\right) +\frac{10lq^{2k}}{mk\left( 2\ln \left( \frac{q}{p}\right) \right) }\right. \\&\left. \quad -\,\frac{2(m-l)}{mk\left( 2\ln \left( \frac{q}{p}\right) \right) } (pq)^k+\frac{(m-2l)}{2m}(pq)^k\right] . \end{aligned}$$

Proof

Proof is exchangeable with Theorem 1 using substitution \(\lambda =1-\frac{l}{m}\) and \(\mu =\frac{l}{m}\). \(\square \)

Theorem 2

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\)such that\(f''\in L^1([p,q])\), where\(p,q\in I\)with\(p<q\). If \(|f''|^k\)isquasi-geometrically convex on [p, q] for some fixed conjugate numbers\(j,k \ge 0 \)where\(k>1\)and\(0 \le \mu \le 1/ 2 \le \lambda \le 1\), then the following inequality holds

$$\begin{aligned}&|M_f\left( \lambda ,\mu ,p,q\right) |\le \left( \ln \left( \frac{q}{p}\right) \right) ^2 \\&\quad \times \,\left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}} \\&\quad \times\, \left\{ c_5^{\frac{1}{j}}(j,\mu )c_7^{\frac{1}{k}}(k,p, q)+c_6^{\frac{1}{j}}(j,\lambda )c_8^{\frac{1}{k}}(k,p,q)\right\} , \end{aligned}$$
(2.5)

where

$$\begin{aligned} c_5(j,\mu ) = \frac{(1-2\mu )^{j+1}}{4^{j+1}(j+1)(1-\mu )}, c_6(j,\lambda ) = \frac{(2\lambda -1)^{j+1}}{4^{j+1}(j+1)\lambda },\\ c_7(k,p,q) = \frac{p^k}{2}\mathrm {L}\left( p^k,q^k\right) , c_8(k,p,q) =\frac{q^k}{2}\mathrm {L}\left( p^k, q^k\right) . \end{aligned}$$

Proof

From Lemma 2 by applying Hölder inequality and using the quasi-geometrically convexity on [p, q] of \(|f''|^k\), we have

$$\begin{aligned}&\left| M_f(\lambda , \mu ,p,q)\right| =\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times\, \left\{ \left( \int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^jd\tau \right) ^{\frac{1}{j}}\left( \int _0^{\frac{1}{2}}\left( p^{2(1-\tau )}q^{2\tau }\right) ^k \right. \right. \\&\quad \left. \left. \times\, \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} d\tau \right) ^{\frac{1}{k}} \right. \\&\quad \left. +\,\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|^jd\tau \right) ^{\frac{1}{j}}\left( \int _{\frac{1}{2}}^1\left( j^{2(1-\tau )}q^{2\tau }\right) ^k\right. \right. \\&\quad \left. \left. \times\, \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} d\tau \right) ^{\frac{1}{k}}\right\} \end{aligned}$$
(2.6)
$$\begin{aligned}&\left| M_f(\lambda ,\mu ,p,q)\right| =\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times \,\left( \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^q\right\} \right) ^{\frac{1}{k}}\left\{ \left( \int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^jd\tau \right) ^{\frac{1}{j}}\right. \\&\quad \left. \left( \int _0^{\frac{1}{2}}\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau \right) ^{\frac{1}{k}}\right. \\&\quad \left. +\,\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|^jd\tau \right) ^{\frac{1}{j}} \left( \int _{\frac{1}{2}}^1\left( p^{2(1-\tau )}k^{2\tau }\right) ^kd\tau \right) ^{\frac{1}{k}}\right\} \end{aligned}$$
(2.7)

where

$$\begin{aligned}&c_5(j,\mu )=\int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^jd\tau =\int _0^\mu \tau ^j (\mu -\tau )^jd\tau \\&\quad +\,\int _\mu ^{\frac{1}{2}} \tau ^j (\tau -\mu )^jd\tau = \frac{(1-2\mu )^{j+1}}{4^{j+1}(j+1)(1-\mu )},\\&c_6(j,\mu )=\int _{\frac{1}{2}}^1|(1-\tau )(\lambda -\tau )|^jd\tau =\int _{\frac{1}{2}}^\lambda (1-\tau )^j (\tau -\lambda )^jd\tau \\&\quad +\,\int _\lambda ^1 (1-\tau )^j(\tau -\lambda )^jd\tau = \frac{(2\lambda -1)^{j+1}}{4^{j+1}(j+1)\lambda }. \end{aligned}$$

In order to calculate \(c_7^{\frac{1}{k}}(k,p,q)\) and \(c_8^{\frac{1}{k}}(k,p,q)\) using substitution \(\xi =p^{2(1-\tau )}q^{2\tau }\), it leads to

$$\begin{aligned}&c_7(k,p,q)=\int _0^{\frac{1}{2}}\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau =\frac{p^k}{2}\mathrm {L}\left( p^k, q^k\right) ,\\&c_8(k,p,q)=\int _{\frac{1}{2}}^1\left( p^{2(1-\tau )}q^{2\tau }\right) ^kd\tau =\frac{q^k}{2}\mathrm {L}\left( p^k, q^k\right) . \end{aligned}$$

Hence, (2.5) easily found from (2.7). \(\square \)

Corollary 2

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\)such that\(f''\in L^1([p,q])\), where\(p,q \in I\)with\(p<q\). If\(|f''|^k\)isquasi-geometrically convex on [p, q] for some fixed conjugate numbers\(j,k \ge 0 \)with\(k > 1\)and for\(l, m \in {\mathbb {R}}\)with\(l < m\), then the following inequality holds

$$\begin{aligned}&\left| m_f\left( \frac{l}{m},p, q\right) \right| \le \left( \ln \left( \frac{q}{p}\right) \right) ^2 \frac{1}{m^j(j+1)} \\&\quad \times \,\left[ \frac{(m-2l)^{j+1}}{4^{j+1}(m-l)}\right] \left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}} \\&\quad \times \,\left\{ c_7^{\frac{1}{k}}(k,p, q)+c_8^{\frac{1}{k}}(k,p,q)\right\} , \end{aligned}$$
(2.8)

where

$$\begin{aligned}&c_5\left( j,\frac{l}{m}\right) =c_6\left( j,\frac{l}{m}\right) =\frac{1}{m^j(j+1)}\left[ \frac{(m-2l)^{j+1}}{4^{j+1}(m-l)}\right] ,\\&c_7(k,p,q) = \frac{p^k}{2}\mathrm {L}\left( p^k,q^k\right) , c_8(k,p,q) =\frac{q^k}{2}\mathrm {L}\left( p^k,q^k\right) , \end{aligned}$$

and\(m_f\left( \frac{l}{m},p,q\right) \)fixed in Corollary 1.

Proof

Proof is exchangeable with Theorem 2 using substitution \(\lambda =1-\frac{l}{m}\) and \(\mu =\frac{l}{m}\). \(\square \)

Theorem 3

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\)such that\(f''\in L^1([p,q])\), where\(p,q\in I\)with\(p<q\). If\(|f''|^k\)isquasi-geometrically convex on [p, q] for some fixed conjugate numbers\(j,k\ge 0 \)where\(k>1\)and\(0 \le \mu \le 1/ 2 \le \lambda \le 1\), then the following inequality holds

$$\begin{aligned}&|M_f\left( \lambda ,\mu ,p,q\right) |\le \left( \ln \left( \frac{q}{p}\right) \right) ^2\left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}} \\&\times \,\left\{ c_7^{\frac{1}{j}}(j,p,q)c_5^{\frac{1}{k}}(k,\mu )+c_8^{\frac{1}{j}}(j,p,q)c_6^{\frac{1}{k}}(k, \lambda )\right\} , \end{aligned}$$
(2.9)

where

$$\begin{aligned}&c_5(k,\mu ) = \frac{(1-2\mu )^{k+1}}{4^{k+1}(k+1)(1-\mu )},c_6(k,\lambda ) = \frac{(2\lambda -1)^{k+1}}{4^{k+1}(k+1)\lambda },\\&c_7(j,p,q) = \frac{p^j}{2}\mathrm {L}\left( p^j,q^j\right) ,c_8(j,p,q) =\frac{q^j}{2}\mathrm {L}\left( p^j,q^j\right) , \end{aligned}$$

such that\(\frac{1}{j}+\frac{1}{k}=1\).

Proof

From Lemma 2 by applying Hölder inequality and using the quasi-geometrically convexity on [p, q] of \(|f''|^k\), we have

$$\begin{aligned}&\left| M_f(\lambda , \mu ,p,q)\right| =\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times\, \left\{ \left( \int _0^{\frac{1}{2}}\left( p^{2(1-\tau )}q^{2\tau }\right) ^jd\tau \right) ^{\frac{1}{j}}\left( \int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^k \right. \right. \\&\quad \left. \left. \times \,\sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} d\tau \right) ^{\frac{1}{k}} \right. \\&\quad \left. +\,\left( \int _{\frac{1}{2}}^1\left( p^{2(1-\tau )}q^{2\tau }\right) ^jd\tau \right) ^{\frac{1}{j}}\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|^k\right. \right. \\&\quad \left. \left. \times\, \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} d\tau \right) ^{\frac{1}{k}}\right\} \end{aligned}$$
(2.10)
$$\begin{aligned}&\left| M_f(\lambda ,\mu ,p,q)\right| =\left( ln\left( \frac{q}{p}\right) \right) ^2 \\&\quad \times \,\left( \sup \left\{ \left| f''(p)\right| ^k, \left| f''(q)\right| ^k\right\} \right) ^{\frac{1}{k}} \\&\quad \left\{ \left( \int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^kd\tau \right) ^{\frac{1}{k}}\right. \\&\quad \left. \times \,\left( \int _0^{\frac{1}{2}}\left( a^{2(1-\tau )}q^{2\tau }\right) ^jd\tau \right) ^{\frac{1}{j}}+\left( \int _{\frac{1}{2}}^1|(1-\tau )(\tau -\lambda )|^kd\tau \right) ^{\frac{1}{k}} \right. \\&\quad \left. \left( \int _{\frac{1}{2}}^1\left( p^{2(1-\tau )}q^{2\tau }\right) ^jd\tau \right) ^{\frac{1}{j}}\right\} ~ \end{aligned}$$
(2.11)

where

$$\begin{aligned}&c_5(k,\mu )=\int _0^{\frac{1}{2}}|\tau (\mu -\tau )|^kd\tau = \left[ \frac{(1-2\mu )^{k+1}}{4^{k+1}(k+1)(1-\mu )}\right] ,\\&c_6(j,\lambda )=\int _{\frac{1}{2}}^1|(1-\tau )(\lambda -\tau )|^kd\tau =\frac{(2\lambda -1)^{k+1}}{4^{k+1}(k+1)\lambda }. \end{aligned}$$

In order to calculate \(c_7^{\frac{1}{k}}(k,p,q)\) and \(c_8^{\frac{1}{k}}(k,p,q)\) using \(\xi =p^{2(1-\tau )}q^{2\tau }\), it leads to

$$\begin{aligned}&c_7(j,p,q)=\int _0^{\frac{1}{2}}\left( p^{2(1-\tau )}q^{2\tau }\right) ^jd\tau =\frac{p^j}{2}\mathrm {L}\left( p^j,q^j\right) ,\\&c_8(j,p,q)=\int _{\frac{1}{2}}^1\left( p^{2(1-\tau )}q^{2\tau }\right) ^jd\tau =\frac{q^j}{2}\mathrm {L}\left( p^j,q^j\right) . \end{aligned}$$

Hence, (2.9) easily found from (2.11). \(\square \)

Corollary 3

A function\(f: I \subseteq {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\)be a twice differentiable function on\(I^o\)such that\(f''\in L^1([p,q])\), where\(p,q \in I\)with\(p<q\). If\(|f''|^k\)isquasi-geometrically convex on [p, q] for some fixed conjugate numbers\(j,k \ge 0 \)and for\(l, m \in {\mathbb {R}}\)with\(l<m\), then the following inequality holds

$$\begin{aligned}&\left| m_f\left( \frac{l}{m},p, q\right) \right| \le \left( \ln \left( \frac{q}{p}\right) \right) ^2\frac{1}{m^k(k+1)} \\&\quad \times \,\left[ \frac{(m-2l)^{k+1}}{4^{k+1}(m-l)}\right] \left( \sup \left\{ |f''(p)|^k, |f''(q)|^k\right\} \right) ^{\frac{1}{k}} \\&\quad \times \,\left\{ c_7^{\frac{1}{j}}(j,p,q) +c_8^{\frac{1}{j}}(j,p,q)\right\} , \end{aligned}$$
(2.12)

where

$$\begin{aligned}&c_5\left( k,\frac{l}{m}\right) =c_6\left( k,\frac{l}{m}\right) =\frac{1}{m^k(k+1)}\left[ \frac{(m-2l)^{k+1}}{4^{k+1}(m-l)}\right] ,\\&\times \,c_7(j,p,q) = \frac{a^p}{2}\mathrm {L}\left( p^j,q^j\right) ,c_8(j,p,q) =\frac{q^j}{2}\mathrm {L}\left( p^j,q^j\right) , \end{aligned}$$

and \(m_f\left( \frac{l}{m},p,q\right) \) fixed in Corollary 1.

Proof

Proof is exchangeable with Theorem 3 using substitution \(\lambda =1-\frac{l}{m}\) and \(\mu =\frac{l}{m}\). \(\square \)

Concluding remarks

Some results have been developed by generalizing both Hadamard’s and Simpson’s inequalities for quasi-geometrically convex mapping by defining a new identity for twice differentiable mappings. It is also to be mentioned that the proofs of the corollaries 1, 2 and 3 lead to respective Theorems 1, 2 and 3 using the substitution \(\lambda =1-\frac{l}{m}\) and \(\mu =\frac{l}{m}\).