An Extention of the Hadamard-Type Inequality for a Convex Function Defined on Modulus of Complex Integral Functions

Md Mainul Islam,A.N.M Rezaul Karim

doi:10.5539/jmr.v5n3p92

Md Mainul Islam, A.N.M Rezaul Karim

Open Access

https://doi.org/10.5539/jmr.v5n3p92

Copy DOI

Abstract

In this paper we extend the Hadamard's type inequalities for convex functions defined on the modulus of integral functions in complex field. Firstly, by using the Principal of maximum modulus theorem we show that $M(r)$ and $lnM(r)$ are continuous and convex functions for any non-negative values of $r$. Finally we derive two inequalities analogous to well known Hadamard's inequality by using elementary analysis.

Highlights

Let f : I ⊆ R → R is a convex mapping defined on the interval I ⊂ R
In this paper we extend the Hadamard’s type inequalities for convex functions defined on the modulus of integral functions in complex field
The main purpose of this paper is to establish some integral inequality involving the modulus of complex integral functions

Summary

Introduction

Let f : I ⊆ R → R is a convex mapping defined on the interval I ⊂ R. Theorem 1.6 (The Principal of Maximum Modulus Theorem) Let f (z) is an analytic function, regular in a region D and on its boundary C, where C is a simple closed contour. Theorem 1.7 (The Principal of Minimum Modulus Theorem) If f (z) is a non-constant integral function without zeros within the region bounded by a closed contour C, | f (z)| obtained its minimum value at a point on the boundary of C, i.e. if m is the minimum value of | f (z)| on C, the inequality holds | f (z)| ≥ m, for any z lies inside C.

Some Useful Lemmas

Main Results

Conclusion