A New Version of q-Hermite-Hadamard’s Midpoint and Trapezoid Type Inequalities for Convex Functions

([26]). Let ϒ : [λ₁, λ₂] → ℝ be a continuous function. Then the left q-derivative of function ϒ at x ∈ [λ₁, λ₂] is defined by

The function ϒ is said to be q-differentiable function on [λ₁, λ₂] if _λ1 D_q ϒ(x) exists for all x ∈ [λ₁, λ₂].

([26]). If ϒ, g: J → ℝ are q-differentiable functions, then the following identities hold:

1. The product ϒ g: [λ₁, λ₂] → ℝ is q-differentiable on [λ₁, λ₂] with

   λ1Dq(Υg)(x)=Υ(x)λ1Dqg(x)+g(qx+(1−q)λ1)λ1Dq(x)=g(x)λ1DqΥ(x)+Υ(qx+(1−q)λ1)λ1Dqg(x)

2. If g(x)g(qx+(1−q)λ1)≠0, then ϒ/g is q-differentiable on [λ₁, λ₂] with

   λ1Dq(Υg)(x)=g(x)λ1DqΥ(x)−Υ(x)λ1Dqg(x)g(x)g(qx+(1−q)λ1)

([26]). Let ϒ : [λ₁, λ₂] → ℝ be a continuous function. Then the left q-integral of function ϒ at z ∈ [λ₁, λ₂] is defined by

The function ϒ is said to be q-integrable function on [λ₁, λ₂] if ∫λ1zΥ(x)λ1dqx exists for all z ∈ [λ₁, λ₂].

([26]). If ϒ : [λ₁, λ₂] → ℝ is a continuous function and z ∈ [λ₁, λ₂], then the following identities hold:

1. λ1Dq∫λ1zΥ(x)λ1dqx=Υ(z);

2. ∫cz λ1Dqϒ(x)λ1dqx=ϒ(z)−ϒ(c) for c∈(λ1,z).

([8]). The right q-derivative of mapping ϒ : [λ₁, λ₂] → ℝ is defined as:

If x = λ₂, we define λ2DqΥ(λ2)=limx→λ2 λ2DqΥ(x) if it exists and it is finite.

([8]). The right q-definite integral of mapping ϒ : [λ₁, λ₂] → ℝ on [λ₁, λ₂] is defined as:

([8]). If ϒ : [λ₁, λ₂] → ℝ is a continuous function and z ∈ [λ₁, λ₂], then the following identities hold:

1. λ1Dq∫zλ2Υ(x)λ1dqx=−Υ(z);

2. ∫cλ2 λ1Dqϒ(x)λ1dqx=ϒ(λ2)−ϒ(z).

([24]). For continuous functions ϒ, g: [λ₁, λ₂] → ℝ, the following equality holds:

([25]). For continuous functions ϒ, g: [λ₁, λ₂] → ℝ, the following equality holds:

Let ϒ: [λ₁, λ₂] → ℝ be a convex mapping, then we have the following inequality:

Since ϒ is a convex function, therefore

By letting x=t2λ1+2−t2λ2 and y=2−t2λ1+t2λ2, we obtain

q-integrating the inequality (3.2) over [0,1], we have

hence, the first inequality proved. We again use the convexity of ϒ to prove the second inequality, we have

q-integrating the inequality (3.3) over [0,1], we have

Thus, we obtain the required inequality.

In Theorem 3.1, if we set q → 1⁻, then we obtain the classical Hermite-Hadamard inequality (1.1).

For a convex function ϒ (x) = x². From Theorem 3.1 with λ₁ = 0, λ₂ = 1 and q=12, we have

and

Thus, it is clear that the obtained inequality in Theorem 3.1 is valid.

Let ϒ: [λ₁, λ₂] ⊂ ℝ → ℝ be a function. If the functions _λ1 D_q ϒ and ^λ₁ D_q ϒ are continuous and integrable over [λ₁, λ₂], then we have the following new equality:

Let

From Lemma 2.1, we have

Similarly, from Lemma 2.2, we obtain the following equality:

Thus, we obtain the resultant equality by putting (4.3) and (4.4) in (4.2). The proof is completed. □

In Lemma 4.1, if we take the limit as q → 1⁻, then we obtain [23: Corollary 1].

We assume that the conditions of Lemma 4.1 hold. If the functions │_λ1 D_q ϒ│ and │^λ₂ D_q ϒ│ are convex, then the following inequality holds:

On taking modulus in (4.1) and using convexity of │_λ1 D_q ϒ│ and │^λ₂ D_q ϒ│, we have

Thus, the proof is completed.

In Theorem 4.1, if we set the limit as q → 1⁻, then we obtain [19: Theorem 2.2].

We assume that the conditions of Lemma 4.1 hold. If the functions │_λ1 D_q ϒ│^s and │^λ₂ D_q ϒ│^s, s ≥ 1 are convex, then the following inequality holds:

By taking modulus in (4.1) and using the power mean inequality, we have

Applying convexity of │_λ1 D_q ϒ│^s and │^λ₂ D_q ϒ│^s, we have

Thus, the proof is completed. □

We assume that the conditions of Lemma 4.1 hold. If the functions │_λ1 D_q ϒ│^s and │^λ₂ D_q ϒ│^s, s > 1 are convex, then the following inequality holds:

where s⁻¹ + r⁻¹ = 1.

On taking modulus in (4.1) and applying Hölder’s inequality, we have

Using convexity of │_λ1 D_q ϒ│^s and │^λ₂ D_q ϒ│^s, we have