Strongly convex functions and extensions of related inequalities with applications to entropy

1.   Introduction

For $a, b\in \mathbb{R}, a < b$, and $I: = [a, b]$, the function $\varphi:I\rightarrow \mathbb{R}$ is convex if

and $\varphi$ is named strongly convex (S-C) with modulus $k$ if

for all $x_1, x_2\in I$ and all $t\in [0, 1]$

Mercer inequality ^[8], Hermite-Hadamard inequality (H-H inequality) ^[6], and Jensen's inequality ^[7] are some types of important inequalities in different fields of mathematical analysis and optimization.

In ^[4], Fejér investigated an extension of the H-H inequality. Azócar et al. ^[1] obtained the following Fejér inequality for the S-C function.

Because of the comprehensive results of the Jensen's, Mercer, and H-H inequalities, some researchers exteneded their studies via mappings of various types (see ^{[2,3,10,17,18]}).

The theory of convex function is significant in entropy estimation and optimization ^{[11,12,13,14,15,16]}.

An equivalent condition for the convexity of a continuous function is given in ^[21]. Also, in ^[22], S. Zlobec generalized some of the basic integral properties of convex functions.

Throughout this article, suppose that $\mathbf{x} = \{x_{i}\}\subseteq [a, b]$ and $\mathbf{t} = \{t_{i}\}, 0\leq t_i$ with $\sum_{i = 1}^{n}t_{i} = 1$.

Theorem 1.1. ^[9] Suppose that $\varphi :I\rightarrow \mathbb{R}$ is an S-C function with modulus $k$, then

for all $x_1, x_2, \dots, x_n\in [a, b], t_{i}\geq 0 (i = 1, \dots, n)$ with $\sum_{i = 1}^{n}t_{i} = 1, \bar{{\bf x}} = \sum_{i = 1}^{n}t_{i}x_{i}$, and $V({\bf x}): = \sum_{i = 1}^{n}t_{i}(x_{i}-\bar{{\bf x}})^2$.

Definition 1.1. The Shannon entropy of a probability distribution $\mathbf{t}$ is defined by

Proposition 1.1. ^[19] Let $\eta: = \min\{t_i: i = 1, \dots, n\}$ and $\vartheta: = \max\{t_i: i = 1, \dots, n\}$, then

Our main goal is to obtain some intersting Jensen, Mercer, and Fejér-Hermite-Hadamard inequalities for S-C functions. Furthermore, we use those inequalities in mathematical analysis and the Shannon entropy to obtain a strong bound for entropy of a probability distribution.

2.   Main results

In this section, we generelize the Mercer, Jensen type, and Fejér-Hermite-Hadamard inequalities for S-C functions.

Lemma 2.1. Suppose that $\varphi :I\rightarrow \mathbb{R}$ is an S-C function with modulus $k$. If $w_{1}, w_{2}, w_{3}\in I$, and $w_1 < w_2 < w_3$, then

$(i)$ $\frac{\varphi(w_2)-\varphi(w_1)}{2}\leq \varphi\left(\frac{w_2+w_3}{2}\right)-\varphi\left(\frac{w_1+w_3}{2}\right)-\frac{k}{4}\left(w_2-w_1\right)\left(2w_3-w_2-w_1\right)$,

$(ii)$ $\frac{\varphi(w_3)-\varphi(w_2)}{2}\geq \varphi\left(\frac{w_1+w_3}{2}\right)-\varphi\left(\frac{w_1+w_2}{2}\right)+\frac{k}{4}\left(w_3+w_2-2w_1\right)\left(w_3-w_2\right)$.

Proof. Since $w_1 < w_2 < \frac{w_2+w_3}{2} < w_3$, there are $s, t\in [0, 1]$, $s+t = 1$ such that $w_2 = s\left(\frac{w_2+w_3}{2}\right)+tw_1.$ Therefore,

Similarly, we obtain (ⅱ) by putting $w_2 = t\left(\frac{w_1+w_2}{2}\right)+sw_3$, where $w_1 < \frac{w_1+w_2}{2} < w_2 < w_3$. □

Theorem 2.1. Assume that $\varphi :[a, b]\rightarrow \mathbb{R}$ is an S-C function with modulus $k$ and

where $a\leq p, q\leq b$, then

Proof. Taking $w_1 = a, w_2 = p$, and $w_3 = b$ in the part (ⅰ) of Lemma 2.1 and $w_1 = p, w_2 = q$, and $w_3 = b$ in the part (ⅱ) of Lemma 2.1, we gain

respectively.

By (2.3), we get

Now, from (2.2) and (2.4), we conclude (2.1). □

Corollary 2.1. Assume that $\varphi :[a, b]\rightarrow \mathbb{R}$ is an S-C function with modulus $k$ and $x\in [a, b]$, then

Proof. Replacing $p$ by $x$ and $q$ by $a+b-x$ in Theorem 2.1 gives the desired result. □

Lemma 2.2. Let $\lambda, \mu\geq 0$ and $\lambda +\mu = 1$ and let $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$, then

for all $x_1, x_2, \dots, x_n\in [a, b], t_{i}\geq 0\; (1\leq i\leq n)$ with $\sum_{i = 1}^{n}t_{i} = 1$ and $\bar{{\bf x}} = \sum_{i = 1}^{n}t_{i}x_{i}$.

Proof. By applying Theorem 1.1 and (2.5), we obtain

□

Corollary 2.2. Let $\lambda, \mu\in [0, 1]$ with $\lambda+\mu = 1$, and let $\varphi$ be an S-C function with modulus $k$ on $[a, b]$ and $x\in [a, b]$, then

Corollary 2.3. Let $\varphi$ be an S-C function with modulus $k$ on $[a, b]$ and $s, t, \lambda, \mu\in [0, 1]$ with $s+t = 1$ and $\lambda+\mu = 1$, then

Proof. Substitute $x$ by $sa+tb$ in Corollary 2.2 to get the inequality. □

Corollary 2.4. Let $\varphi$ be an S-C function with modulus $k$ on $[a, b]$ and $s, t\in [0, 1], s+t = 1$, then

Proof. Set $\lambda = 1$ in Corollary 2.3 to get the righthand side of the inequality. On the other hand, by definition, we have

□

The Corollary 2.4 shows a kind of pre-H-H inequalities.

Next, we prove a generalization of the H-H type inequality for S-C functions.

Theorem 2.2. Assume that $\varphi :[a, b]\rightarrow \mathbb{R}$ is an S-C function with modulus $k$ and $\omega$ is a nonnegative function on $[a, b]$, then

Proof. By multiplying the two-sided inequality in Corollary 2.4 with $\omega(a+t(b-a))$, integrating with respect to variable $t$ from $0$ to $1$, and by interchanging $u = a+t(b-a)$, we get

since

□

Corollary 2.5. Let $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$, then

Proof. Putting $\omega\equiv 1$ in Theorem 2.2 and after some calculations, the desired inequality follows. □

If we consider $\omega(u) = u^{\alpha -1}$ in Theorem 2.2, then we have the following result.

Corollary 2.6. Let $\alpha > 0$, $0 < a < b$, and $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$, then

If $\alpha\rightarrow 0^+$ in Corollary 2.6, then we get the next corollary.

Corollary 2.7. Let $0 < a < b$ and $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$, then

Let $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$ and $0\leq s, t\leq 1$ such that $s+t = 1$. We define

for all $x, y\in [a, b]$.

Theorem 2.3. Let $\varphi :[a, b]\rightarrow \mathbb{R}$ be an S-C function with modulus $k$, then

Proof. First, we prove that

for all $s, t\in \mathbf{t}$ and $x, y\in [a, b]$.

Now,

The remains of the proof is an application of Theorem 2.1. □

Theorem 2.4. Suppose that $\varphi :[a, b]\rightarrow \mathbb{R}$ is an S-C function with modulus $k$, then

where $(b-x_r)(x_r-a) = \min_{i}\left\{(b-x_i)(x_i-a)\right\}$.

Proof. Since $x_i\in[a, b]$, there is a sequence $\{\lambda_{i}\}, \lambda_{i}\in[0, 1]$ with $x_i = \lambda_{i}a+(1-\lambda_{i})b$ for $i = 1, 2, \dots$. Thus,

Setting $\lambda: = \sum_{i = 1}^{n}t_{i} \lambda_{i}$ and $\mu : = 1-\sum_{i = 1}^{n}t_{i} \lambda_{i}$, by the use of Theorem 2.3, we gain

because

for all $i = 1, 2, \dots, n$. Hence,

□

3.   Applications

3.1.   Applications in analysis

In the following, we obtain an application of Lemma 2.2, which improves the results from ^[8].

Proposition 3.1. Let $0 < a\leq x_{i}\leq b$, $t_{i}\geq 0 \; (i = 1, 2, \dots, n)$ with $\sum_{i = 1}^{n}t_{i} = 1, \lambda, \mu\geq 0$, and $\lambda +\mu = 1$, then

where $\tilde{A}_{\lambda}: = \lambda (a+b)+(\mu-\lambda)\bar{{\bf x}}, \tilde{G}_{\lambda}: = \frac{(ab)^\lambda}{\prod_{i = 1}^{n}x_{i}^{t_{i}(\lambda - \mu)}}, y_{i}: = (a+b-2x_{i})^2$, and ${\bf y}: = \{y_i\}$.

Proof. Letting $\varphi(x): = -\log(x)$ and $k = \frac{1}{2b^2}$ in Lemma 2.2, the desired inequality follows. □

Remark 3.1. Putting $\lambda = 1$ in (3.1), we obtain

where $\tilde{A} = a+b-\bar{{\bf x}}$ and $\tilde{G} = \frac{ab}{\prod_{i = 1}^{n}x_{i}^{t_{i}}}$ (see ^[8]).

Example 3.1. Assume that $\beta \geq 2$ and $0 < a < b$, then $\varphi(x) = x^{\beta}$ is an S-C function with modulus $k: = \frac{\beta(\beta-1)}{2}a^{\beta -2}$ on $[a, b]$. Further, if $\beta = 2n \; (n = 1, 2, \dots)$, then $\varphi(x) = x^{\beta}$ is an S-C function with modulus $k$ on arbitrary interval $[a, b]$.

Proof. Let $x, y\in [a, b]$. Define the following functions on $[0, 1]$:

and

Since $g(0) = h(0)$, $g(1) = h(1)$, $h^{\prime\prime}\equiv 0$, and

$g(t)\leq h(t)$ for every $t\in [0, 1]$. Therefore, $x^{\beta}$ is an S-C function with modulus $\frac{\beta(\beta-1)}{2}a^{\beta -2}$ on $[a, b]$. □

In the next proposition, we give an extension of the pre-Grüss inequality (see ^[5,20]):

Proposition 3.2. Assume that $x_{i}\in [a, b]$ and $\{t_i\}\in {\bf t}$, then

where $(b-x_r)(x_r-a) = \min_{i}\left\{(b-x_i)(x_i-a)\right\}$.

Proof. It follows from Example 3.1 and Theorem 2.4 with $\varphi(x) = x^2$. □

3.2.   Applications to entropy

In this subsection, new Shannon entropy bounds are found, that improve the entropy bounds from ^[19].

Proposition 3.3. Assume that $\eta: = \min\{t_i: i = 1, \dots, n\}$ and $\vartheta: = \max\{t_i: i = 1, \dots, n\}$, then

Proof. Replace $\varphi(x)$ by $-\log(x)$ in Theorem 2.4 and consider $k = \frac{1}{2b^2}, x_i: = \frac{1}{t_i}$ for all $i = 1, \ldots, n$. □

Proposition 3.4. Assume that $\varphi :[a, b]\rightarrow \mathbb{R}$ is an S-C function with modulus $k$, then

where $(b-x_r)(x_r-a) = \min_{i}\left\{(b-x_i)(x_i-a)\right\}$.

Proof. Set $t_i = \frac{1}{n}$ for every $i = 1, \ldots, n$ in Theorem 2.4. □

Proposition 3.5. Assume that $\eta: = \min\{t_i: i = 1, \dots, n\}$ and $\vartheta: = \max\{t_i: i = 1, \dots, n\}$, then

Proof. Replace $\varphi(x)$ by $x\log(x)$ in Proposition 3.4 and modulus $k = \frac{1}{2b}, x_i: = t_i$ for all $i = 1, \ldots, n$. □

Example 3.2. Let $j\geq 2$ be an integer, $n = 10^j$, $\eta = 10^{-j-1}$, and $\vartheta = 10^{-j+1}$, then

and

4.   Conclusions

In this work, we have stablished some new inequalities such as the Mercer, Fejér-Hermite-Hadamard, and Jensen inequalities for strongly convex functions. Next, using these inequalities, we get some applications in analysis and entropy of probability distributions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no competing interests.