Convex functions and their generalizations play a significant role in scientific observation and calculation of various parameters in modern analysis, especially in the theory of optimization. Moreover, convex functions have some nice properties, such as differentiability, monotonicity, and continuity, which are useful in applications [1–5]. Interest in mathematical inequalities for convex and generalized convex functions has been growing exponentially, and research in this respect has had a significant impact on modern analysis [6–20]. Several mathematical inequalities have been established for s-convex functions in particular [21–28], one of the most important being the Jensen inequality. In this paper, we study the Jensen inequality in a more standard framework for s-convex functions.

Definition 1.1 (s-convexity [29]). For s > 0 and a convex subset B of a real linear space S, a function Γ : B → ℝ is said to be s-convex if the inequality

holds for all ε₁, ε₂ ∈ B and κ₁, κ₂ ≥ 0 with κ₁ + κ₂ = 1.

The function Γ is said to be s-concave if the inequality (1.1) holds in the reverse direction. Obviously, for s = 1 an s-convex function becomes a convex function, which shows that s-convexity of a function is a generalization of ordinary convexity of that function.

Lemma 1.2 ([29]). Let B be a convex subset of a real linear space S and let Γ : B → ℝ be a convex function. Then the following two statements hold:

The Green function [30]

defined on [α₁, α₂] × [α₁, α₂] and the integral identity

for the function Γ∈C2[α1,α2] will be used to obtain the main results. Note that G₁ is convex and continuous with respect to both variables.

This paper is organized as follows. In section 2 we give a bound for the Jensen gap in discrete form, which pertains to functions for which the absolute value of the second derivative is s-convex. We also derive a bound for the integral version of the Jensen gap. Then we conduct two numerical experiments that provide evidence for the tightness of the bound in the main result. We deduce a converse of the Hölder inequality from the discrete result and a bound for the Hermite-Hadamard gap from the integral result. Moreover, as a consequence of the integral result we obtain a converse of the Hölder inequality in its corresponding integral version. At the beginning of section 3 we present bounds for the Csiszár and Rényi divergences in the discrete case. Finally, we give estimates for the Shannon entropy, Kullback-Leibler divergence, χ² divergence, Bhattacharyya coefficient, Hellinger distance, and triangular discrimination as applications of the bound obtained for the Csiszár divergence. Conclusions are presented in the final section.

Using the concept of s-convexity, we derive a bound for the Jensen gap in discrete form, which is presented in the following theorem.

Theorem 2.1. Suppose |Γ|″ is s-convex for a function Γ∈C2[α1,α2] and that z_i ∈ [α₁, α₂] and κ_i ∈ [0, ∞) for i = 1, …, n with ∑i=1nκi=K>0. Then the following inequality holds:

Proof: Using (1.3), we get

and

Equations (2.5) and (2.6) give

Taking the absolute value of (2.7), we get

By applying a change of variable x = tα₁ + (1 − t)α₂ for t ∈ [0, 1] and using the convexity of G₁(t, x), the inequality (2.8) is transformed to

where z-=1K∑i=1nκizi. The inequality (2.9) leads to the following by using s-convexity of the function |Γ|″:

Now, by using the change of variable x = tα₁ + (1 − t)α₂ for t ∈ [0, 1], we obtain

Upon replacing z_i by z- in (2.11), we get

Also,

Upon replacing z_i by z- in (2.13), we get

The result (2.4) is then obtained by substituting the values from (2.11)–(2.14) into (2.10).

Remark 2.2. If we use the Green function G₂, G₃, or G₄ instead of G₁ in Theorem 2.1, where G₂, G₃, and G₄ are given in [30], we obtain the same result (2.4).

In the following theorem, we give a bound for the Jensen gap in integral form.

Theorem 2.3. Suppose |Γ″| is an s-convex function for Γ∈C2[α1,α2], and let ξ₁ and ξ₂ be real-valued functions defined on [c₁, c₂] with ξ₁(y) ∈ [α₁, α₂] for all y ∈ [c₁, c₂] and such that ξ₂, ξ₁ξ₂, and (Γ ◦ ξ₁) ξ ₂ are all integrable functions on [c₁, c₂]. Then the inequality

holds provided that ∫c1c2ξ2(y)dy:=ξ>0 when ξ₂(y) ∈ [0, ∞) for all y ∈ [c₁, c₂].

Proof: Using the same procedure as in the proof of Theorem 2.1, (2.15) can be obtained.

Example 1. Let Γ(y)=415y52, ξ1(y)=y2, and ξ₂(y) = 1 for all y ∈ [0, 1]. Then Γ″(y)=y12>0 for all y ∈ [0, 1]. This shows that Γ is a convex function while |Γ″| is 12-convex. Also, ξ₁(y) ∈ [0, 1] for all y ∈ [0, 1] and we have [α₁, α₂] = [c₁, c₂] = [0, 1]. Now, the left-hand side of inequality (2.15) gives ∫01Γ(ξ1(y))dy-Γ(∫01ξ1(y)dy)=0.0444-0.0171=0.0273=E1, which shows how sharp the Jensen inequality is. The right-hand side of (2.15) gives 0.0274, which is very close to the true discrepancy E₁. That is, from inequality (2.15) we have

The difference 0.0274 − 0.0273 = 0.0001 between the two sides of (2.16) shows that the bound for the Jensen gap given by inequality (2.15) is very close to the true value.

Example 2. Let Γ(y)=100231y2110, ξ₁(y) = y, and ξ₂(y) = 1 for all y ∈ [0, 1]. Then Γ″(y)=y110>0 for all y ∈ [0, 1], which shows that Γ is a convex function while |Γ″| is s-convex with s=110. Also, ξ₁(y) ∈ [0, 1] for all y ∈ [0, 1] and we have [α₁, α₂] = [c₁, c₂] = [0, 1]. Therefore, from the left-hand side of inequality (2.15) we obtain ∫01Γ(ξ1(y))dy-Γ(∫01ξ1(y)dy)=0.1396-0.1010=0.0386=E2, which shows that the Jensen inequality is quite sharp. The right-hand side of (2.15) gives 0.0387, a value very close to the true discrepancy E₂. Finally, from inequality (2.15) we have

The difference 0.0387 − 0.0386 = 0.0001 between the two sides of (2.17) provides further evidence of the tightness of the bound for the Jensen gap given by inequality (2.15).

As an application of Theorem 2.1, we derive a converse of the Hölder inequality, stated in the following proposition.

Proposition 2.4. Let q₂ > 1 and q₁ ∉ (2, 3) be such that 1q1+1q2=1, and let s ∈ (0, 1]. Also, let [α₁, α₂] be a positive interval and let (d₁, …, d_n) and (b₁, …, b_n) be two positive n-tuples such that ∑i=1ndibi∑i=1nbiq2, with dibi-q2q1∈[α1,α2] for i = 1, …, n. Then

Proof: Let Γ(x)=xq1 for x ∈ [α₁, α₂]; then Γ″(x)=q1(q1-1)xq1-2>0 and |Γ″|″(x)=q1(q1-1)(q1-2)(q1-3)xq1-4>0, which shows that Γ and |Γ″| are convex functions. The function |Γ″| is also non-negative, so by Lemma 1.2 it is also an s-convex function for s ∈ (0, 1]. Thus, using (2.4) with Γ(x)=xq1, κi=biq2, and zi=dibi-q2q1, we derive

By using the inequality x^γ − y^γ ≤ (x − y)^γ for 0 ≤ y ≤ x and γ ∈ [0, 1] with x=(∑i=1ndiq1)×(∑i=1nbiq2)q1-1, y=(∑i=1ndibi)q1, and γ=1q1, we obtain

The inequality (2.18) follows from (2.19) and (2.20).

In the following proposition, we provide a converse of the Hölder inequality in integral form as an application of Theorem 2.3.

Proposition 2.5. Let q₂ > 1 and q₁ ∉ (2, 3) be such that 1q1+1q2=1. Also, let ζ1,ζ2:[c1,c2]→ℝ+ be two functions such that ζ1q1(y), ζ2q2(y), and ζ₁(y)ζ₂(y) are integrable on [c₁, c₂] with ζ1(y)ζ2-q2/q1(y)∈[α1,α2] when [α₁, α₂] ⊂ ℝ. Then the inequality

holds for s ∈ (0, 1].

Proof: Using (2.15) with Γ(x)=xq1 for x∈[α1,α2], ξ2(y)=ζ2q2(y), and ξ1(y)=ζ1(y)ζ2-q2q1(y) and following the procedure of Proposition 2.4, we deduce (2.21).

As an application of Theorem 2.3, in the following corollary we establish a bound for the Hermite-Hadamard gap.

Corollary 2.6. Let ψ∈C2[c1,c2] be a function such that |ψ″| is s-convex; then

Proof: The inequality (2.22) can be obtained by using (2.15) with ψ = Γ, [α₁, α₂] = [c₁, c₂], ξ₂(y) = 1, and ξ₁(y) = y for y ∈ [c₁, c₂].

Definition 3.1 (Csiszár f-divergence [31]). Let t=(t1,…,tn)∈ℝn and r=(r1,…,rn)∈ℝ+n with tiri∈[α1,α2] (i=1,…,n) for [α₁, α₂] ⊂ ℝ. For a function f :[α₁, α₂] → ℝ, the Csiszár f-divergence functional is defined as

Theorem 3.2. Let f∈C2[α1,α2] be a function such that |f″| is s-convex. Then for t=(t1,…,tn)∈ℝn and r=(r1,…,rn)∈ℝ+n the inequality

holds provided that ∑i=1nti∑i=1nri,tiri∈[α1,α2] for i = 1, …, n.

Proof: The inequality (3.23) can easily be deduced from (2.4) by taking Γ=f, zi=tiri, and κi=ri∑i=1nri.

Definition 3.3 (Rényi divergence [31]). For μ ≥ 0 with μ ≠ 1 and two positive probability distributions t = (t₁, …, t_n) and r = (r₁, …, r_n), the Rényi divergence is defined as

Corollary 3.4. Let 0 < s ≤ 1 and [α1,α2]⊆ℝ+. Then for positive probability distributions t = (t₁, …, t_n) and r = (r₁, …, r_n), the inequality

holds provided that ∑i=1nri(tiri)μ,(tiri)μ-1∈[α1,α2] for i = 1, …, n with μ > 1.

Proof: Let Γ(x)=-1μ-1logx for x ∈ [α₁, α₂]. Then Γ″(x)=1(μ-1)x2>0 and |Γ″|″(x)=6(μ-1)x4>0, which shows that Γ and |Γ″| are convex functions with |Γ″| ≥ 0; so by Lemma 1.2 the function |Γ″| is s-convex for s ∈ (0, 1]. Therefore, using (2.4) with Γ(x)=-1μ-1logx, κi=ti, and zi=(tiri)μ-1, we derive (3.24).

Definition 3.5 (Shannon entropy [31]). Let r = (r₁, …, r_n) be a positive probability distribution; then the Shannon entropy is defined as

Corollary 3.6. Let [α1,α2]⊆ℝ+, and let r = (r₁, …, r_n) be a positive probability distribution such that 1ri∈[α1,α2] for i = 1, …, n with 0 < s ≤ 1. Then

Proof: Let f(x) = −log x for x ∈ [α₁, α₂]. Then f″(x)=1x2>0 and |f″|″(x)=6x4>0, which shows that f and |f″| are convex functions. Also, |f″| is non-negative and so by Lemma 1.2 we conclude that it is s-convex for s ∈ (0, 1]. Therefore, using (3.23) with f(x) = −log x and (t₁, …, t_n) = (1, …, 1), we get (3.25).