# Jensen-Type Inequalities for (h, g; m)-Convex Functions

## Abstract

**:**

## 1. Introduction

A function $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex function if

holds for all points x and y in I and all $\lambda \in [0,1]$.

It is called strictly convex if the inequality (1) holds strictly whenever x and y are distinct points and $\lambda \in (0,1)$. If $-f$ is convex (respectively, strictly convex) then we say that f is concave (respectively, strictly concave). If f is both convex and concave, then f is said to be affine.

**Lemma**

**1**

**.**Let ${x}_{1},{x}_{2},{x}_{3}\in I$ be such that ${x}_{1}<{x}_{2}<{x}_{3}$. The function $f:I\to \mathbb{R}$ is convex if and only if the following inequality holds

**Theorem**

**1**

**.**A real-valued function f defined on an interval I is convex if and only if for all ${x}_{1},\dots ,{x}_{n}$ in I and all scalars ${\lambda}_{1},\dots ,{\lambda}_{n}$ in $[0,1]$ with ${\sum}_{i=1}^{n}{\lambda}_{i}=1$ we have

**Definition**

**1**

**.**Let h be a nonnegative function on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, $h\not\equiv 0$ and g be a positive function on $I\subseteq \mathbb{R}$. Let $m\in (0,1]$. A function $f:I\to \mathbb{R}$ is said to be $(h,g;m)$-convex function if it is nonnegative and if

**Lemma**

**2**

**.**If $f:I\to [0,\infty )$ is an $(h,g;m)$-convex function such that $f\left(0\right)=0$, $g\left(x\right)\le 1$ and $h\left(\lambda \right)\le \lambda $, then f is a starshaped, that is $f\left(\lambda x\right)\le \lambda f\left(x\right)$.

**Proposition**

**1**

**.**Let ${h}_{1},{h}_{2}$ be nonnegative functions on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, ${h}_{1},{h}_{2}\not\equiv 0$, such that

**Proposition**

**2**

**.**Let h be a nonnegative function on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, $h\not\equiv 0$ and g be a positive function on $I\subseteq \mathbb{R}$. Let $m\in (0,1]$ and $\alpha >0$. If ${f}_{1},{f}_{2}:I\to [0,\infty )$ are $(h,g;m)$-convex function, then ${f}_{1}+{f}_{2}$ and $\alpha {f}_{1}$ are $(h,g;m)$-convex.

**Proposition**

**3**

**.**Let h be a nonnegative function on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, $h\not\equiv 0$ and g be a positive increasing function on $I\subseteq \mathbb{R}$. Let $0<n<m\le 1$. If $f:I\to [0,\infty )$ is an $(h,g;m)$-convex function such that $f\left(0\right)=0$, $g\left(x\right)\le 1$ and $h\left(\lambda \right)\le \lambda $, then f is $(h,g;n)$-convex.

**Proposition**

**4**

**.**Let ${h}_{1},{h}_{2}$ be nonnegative functions on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, ${h}_{1},{h}_{2}\not\equiv 0$ and let

**Definition**

**2.**

## 2. Schur Type Inequalities

**Proposition**

**5.**

**Proof.**

Recall the Schur inequality:If $x,y,z$ are positive numbers and if λ is real, then

with equality if and only if $x=y=z$.

**Corollary**

**1.**

**Corollary**

**2.**

**Remark**

**1.**

## 3. Jensen-Type Inequalities for $(\mathit{h},\mathit{g};\mathit{m})$-Convex Functions

**Theorem**

**2**

**.**Let ${p}_{1},\dots ,{p}_{n}$ be positive real numbers. Let f be a nonnegative $(h,g;m)$-convex function on $[0,\infty )$ such that $I\subseteq [0,\infty )$, where h is a nonnegative supermultiplicative function on $J\subseteq \mathbb{R}$, $(0,1)\subseteq J$, $h\not\equiv 0$, g is a positive function on $[0,\infty )$ and $m\in (0,1]$. Then, for ${x}_{1},\dots ,{x}_{n}\in I$ the following inequality holds

**Proof.**

**Remark**

**2.**

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

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**MDPI and ACS Style**

Andrić, M.
Jensen-Type Inequalities for (*h*, *g*; *m*)-Convex Functions. *Mathematics* **2021**, *9*, 3312.
https://doi.org/10.3390/math9243312

**AMA Style**

Andrić M.
Jensen-Type Inequalities for (*h*, *g*; *m*)-Convex Functions. *Mathematics*. 2021; 9(24):3312.
https://doi.org/10.3390/math9243312

**Chicago/Turabian Style**

Andrić, Maja.
2021. "Jensen-Type Inequalities for (*h*, *g*; *m*)-Convex Functions" *Mathematics* 9, no. 24: 3312.
https://doi.org/10.3390/math9243312