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

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

De Gruyter | Published online: February 25, 2021

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.

1 Introduction

Let be an interval in R . Then G : 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 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 : 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 : ( 0 , 1 ) R , ϑ = { ϑ ( m ) } m = 0 be a bounded sequence of real numbers and 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) υ 1 , υ 2 ϑ ( z ) = 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 1δ 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:
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, ( ( 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 is said to be a generalized ψ-convex set, if

(2.2) η 1 + ξ υ 1 , υ 2 ϑ ( η 2 η 1 ) K
for all η 1 , η 2 K , 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 R and a mapping G : K R is said to be generalized ψ-convex, if

(2.3) G ( η 1 + ξ 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 , ξ [ 0 , 1 ] .

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

Definition 2.3

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

(2.4) G ( η 1 + ξ 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 , 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 R is said to be generalized ψ-s-convex, if

(2.5) G ( η 1 + ξ 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 , ξ [ 0 , 1 ] .

Definition 2.5

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

(2.6) G ( η 1 + ξ υ 1 , υ 2 ϑ ( η 2 η 1 ) ) sup G ( η 1 ) , G ( η 2 ) for all η 1 , η 2 K , 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 1z), 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 ξζ > 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 1η 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 1η 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 1η 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 1υ 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 + υ 1 , υ 2 ϑ ( υ 2 υ 1 ) ] R R on ϒ (the interior of ϒ) having 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 + υ 1 , υ 2 ϑ ( υ 2 υ 1 ) ] . Then the following equality holds:

(2.15) q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) × η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 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 1υ 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( 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 ( 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 , then we get

q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 1υ 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( 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 ( 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 ( 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 , then we get

q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 1υ 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 1υ 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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

q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 2 1 + q 0 1 ξ 0 d q ξ 1 1 α × 0 1 ξ ( 1 q ξ ) α D q 2 η 1 G ( η 1 + ξ υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 2 1 + q 0 1 ξ 0 d q ξ 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 .
Applying Lemma 2.3, we have
(4.3) q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 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

Let υ 1υ 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 αβ > 1,  and α −1 + β −1 = 1,  then

(4.4) q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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

q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( 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 × sup D q 2 η 1 G ( η 1 ) α , D q 2 η 1 G ( η 2 ) α 0 1 d q 0 ξ 1 / α .
Applying Lemma 2.2, we have
(4.5) q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 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 1υ 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 αβ > 1,  and α −1 + β −1 = 1,  then

(4.6) q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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

q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z = q 2 ( 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 υ 1 , υ 2 ϑ ( η 2 η 1 ) ) d q 0 ξ q 2 ( υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 2 1 + q 0 1 ξ ( 1 q ξ ) β d q 0 ξ 1 β × 0 1 ξ D q 2 η 1 G ( η 1 + ξ υ 1 , υ 2 ϑ ( η 2 η 1 ) ) α d q 0 ξ 1 / α q 2 ( υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 2 1 + q 0 1 ξ ( 1 q ξ ) β 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 0 1 ξ d q 0 ξ 1 / α .
Applying Lemma 2.3, we have
q G ( η 1 ) + G ( η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z q 2 ( 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 + υ 1 , υ 2 ϑ ( η 2 η 1 ) ) 1 + q 1 υ 1 , υ 2 ϑ ( η 2 η 1 ) η 1 η 1 + υ 1 , υ 2 ϑ ( η 2 η 1 ) G ( z ) d q η 1 z