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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 EMAIL logo
Published/Copyright: February 25, 2021
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Open Physics
From the journal Open Physics Volume 19 Issue 1

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

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 0 d q 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 ∫ 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 + 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

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

(4.4) 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

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 ∫ 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 + 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

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 ( 1 − q ξ ) β d q 0 ξ 1 β × ∫ 0 1 ξ 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 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 + 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

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

(4.7) 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

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 d q 0 ξ 1 β × ∫ 0 1 ( 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 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 ( 1 − q ξ ) α d q 0 ξ 1 / α .

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

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 ( 1 − q ξ ) β d q 0 ξ 1 β × ∫ 0 1 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 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 α d q 0 ξ 1 / α .

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

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 ) d q 0 ξ 1 β × ∫ 0 1 ( 1 − q ξ ) 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 ξ β ( 1 − q ξ ) 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 ( 1 − q ξ ) d q 0 ξ 1 / α .

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

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 ( 1 − q ξ ) d q 0 ξ 1 − 1 α × ∫ 0 1 ( 1 − q ξ ) 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 ξ β ( 1 − q ξ ) 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 ( 1 − q ξ ) d q 0 ξ 1 / α .

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

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:

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.

  1. Availability of supporting data: Not applicable. Competing interests: The authors declare that they have no competing interests.

  2. Disclosure: The authors declare that they have no competing interests.

  3. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 61673169).

  4. 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

© 2021 Saima Rashid et al., published by DeGruyter

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

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https://doi.org/10.1515/phys-2021-0001
Keywords for this article
quantum estimates; Hermite–Hadamard inequalities; convex functions; generalized ψ-convex functions; Raina function
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  7. Utilization of updated version of heat flux model for the radiative flow of a non-Newtonian material under Joule heating: OHAM application
  8. Verification of the accommodative responses in viewing an on-axis analog reflection hologram
  9. Irreversibility as thermodynamic time
  10. A self-adaptive prescription dose optimization algorithm for radiotherapy
  11. Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics
  12. The diffusion mechanism of the application of intelligent manufacturing in SMEs model based on cellular automata
  13. Numerical analysis of free convection from a spinning cone with variable wall temperature and pressure work effect using MD-BSQLM
  14. Numerical simulation of hydrodynamic oscillation of side-by-side double-floating-system with a narrow gap in waves
  15. Closed-form solutions for the Schrödinger wave equation with non-solvable potentials: A perturbation approach
  16. Study of dynamic pressure on the packer for deep-water perforation
  17. Ultrafast dephasing in hydrogen-bonded pyridine–water mixtures
  18. Crystallization law of karst water in tunnel drainage system based on DBL theory
  19. Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study
  20. Application of Fibonacci heap to fast marching method
  21. An analytical investigation of the mixed convective Casson fluid flow past a yawed cylinder with heat transfer analysis
  22. Considering the effect of optical attenuation on photon-enhanced thermionic emission converter of the practical structure
  23. Fractal calculation method of friction parameters: Surface morphology and load of galvanized sheet
  24. Charge identification of fragments with the emulsion spectrometer of the FOOT experiment
  25. Quantization of fractional harmonic oscillator using creation and annihilation operators
  26. Scaling law for velocity of domino toppling motion in curved paths
  27. Frequency synchronization detection method based on adaptive frequency standard tracking
  28. Application of common reflection surface (CRS) to velocity variation with azimuth (VVAz) inversion of the relatively narrow azimuth 3D seismic land data
  29. Study on the adaptability of binary flooding in a certain oil field
  30. CompVision: An open-source five-compartmental software for biokinetic simulations
  31. An electrically switchable wideband metamaterial absorber based on graphene at P band
  32. Effect of annealing temperature on the interface state density of n-ZnO nanorod/p-Si heterojunction diodes
  33. A facile fabrication of superhydrophobic and superoleophilic adsorption material 5A zeolite for oil–water separation with potential use in floating oil
  34. Shannon entropy for Feinberg–Horodecki equation and thermal properties of improved Wei potential model
  35. Hopf bifurcation analysis for liquid-filled Gyrostat chaotic system and design of a novel technique to control slosh in spacecrafts
  36. Optical properties of two-dimensional two-electron quantum dot in parabolic confinement
  37. Optical solitons via the collective variable method for the classical and perturbed Chen–Lee–Liu equations
  38. Stratified heat transfer of magneto-tangent hyperbolic bio-nanofluid flow with gyrotactic microorganisms: Keller-Box solution technique
  39. Analysis of the structure and properties of triangular composite light-screen targets
  40. Magnetic charged particles of optical spherical antiferromagnetic model with fractional system
  41. Study on acoustic radiation response characteristics of sound barriers
  42. The tribological properties of single-layer hybrid PTFE/Nomex fabric/phenolic resin composites underwater
  43. Research on maintenance spare parts requirement prediction based on LSTM recurrent neural network
  44. Quantum computing simulation of the hydrogen molecular ground-state energies with limited resources
  45. A DFT study on the molecular properties of synthetic ester under the electric field
  46. Construction of abundant novel analytical solutions of the space–time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods
  47. Some common and dynamic properties of logarithmic Pareto distribution with applications
  48. Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model
  49. Fractional modeling of COVID-19 epidemic model with harmonic mean type incidence rate
  50. Liquid metal-based metamaterial with high-temperature sensitivity: Design and computational study
  51. Biosynthesis and characterization of Saudi propolis-mediated silver nanoparticles and their biological properties
  52. New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation
  53. Modal characteristics of harmonic gear transmission flexspline based on orthogonal design method
  54. Revisiting the Reynolds-averaged Navier–Stokes equations
  55. Time-periodic pulse electroosmotic flow of Jeffreys fluids through a microannulus
  56. Exact wave solutions of the nonlinear Rosenau equation using an analytical method
  57. Computational examination of Jeffrey nanofluid through a stretchable surface employing Tiwari and Das model
  58. Numerical analysis of a single-mode microring resonator on a YAG-on-insulator
  59. Review Articles
  60. Double-layer coating using MHD flow of third-grade fluid with Hall current and heat source/sink
  61. Analysis of aeromagnetic filtering techniques in locating the primary target in sedimentary terrain: A review
  62. Rapid Communications
  63. Nonlinear fitting of multi-compartmental data using Hooke and Jeeves direct search method
  64. Effect of buried depth on thermal performance of a vertical U-tube underground heat exchanger
  65. Knocking characteristics of a high pressure direct injection natural gas engine operating in stratified combustion mode
  66. What dominates heat transfer performance of a double-pipe heat exchanger
  67. Special Issue on Future challenges of advanced computational modeling on nonlinear physical phenomena - Part II
  68. Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation
  69. New quantum integral inequalities for some new classes of generalized ψ-convex functions and their scope in physical systems
  70. Computational fluid dynamic simulations and heat transfer characteristic comparisons of various arc-baffled channels
  71. Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena
  72. Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov–Kuznetsov model
  73. On the optical solutions to nonlinear Schrödinger equation with second-order spatiotemporal dispersion
  74. Analysis of couple stress fluid flow with variable viscosity using two homotopy-based methods
  75. Quantum estimates in two variable forms for Simpson-type inequalities considering generalized Ψ-convex functions with applications
  76. Series solution to fractional contact problem using Caputo’s derivative
  77. Solitary wave solutions of the ionic currents along microtubule dynamical equations via analytical mathematical method
  78. Thermo-viscoelastic orthotropic constraint cylindrical cavity with variable thermal properties heated by laser pulse via the MGT thermoelasticity model
  79. Theoretical and experimental clues to a flux of Doppler transformation energies during processes with energy conservation
  80. On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation
  81. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part II
  82. Numerical study on heat transfer and flow characteristics of nanofluids in a circular tube with trapezoid ribs
  83. Experimental and numerical study of heat transfer and flow characteristics with different placement of the multi-deck display cabinet in supermarket
  84. Thermal-hydraulic performance prediction of two new heat exchangers using RBF based on different DOE
  85. Diesel engine waste heat recovery system comprehensive optimization based on system and heat exchanger simulation
  86. Load forecasting of refrigerated display cabinet based on CEEMD–IPSO–LSTM combined model
  87. Investigation on subcooled flow boiling heat transfer characteristics in ICE-like conditions
  88. Research on materials of solar selective absorption coating based on the first principle
  89. Experimental study on enhancement characteristics of steam/nitrogen condensation inside horizontal multi-start helical channels
  90. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part I
  91. Numerical exploration of thin film flow of MHD pseudo-plastic fluid in fractional space: Utilization of fractional calculus approach
  92. A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation
  93. Stable novel and accurate solitary wave solutions of an integrable equation: Qiao model
  94. Novel soliton solutions to the Atangana–Baleanu fractional system of equations for the ISALWs
  95. On the oscillation of nonlinear delay differential equations and their applications
  96. Abundant stable novel solutions of fractional-order epidemic model along with saturated treatment and disease transmission
  97. Fully Legendre spectral collocation technique for stochastic heat equations
  98. Special Issue on 5th International Conference on Mechanics, Mathematics and Applied Physics (2021)
  99. Residual service life of erbium-modified AM50 magnesium alloy under corrosion and stress environment
  100. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part I
  101. Diverse wave propagation in shallow water waves with the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and Benney–Luke integrable models
  102. Intensification of thermal stratification on dissipative chemically heating fluid with cross-diffusion and magnetic field over a wedge
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