Abstract
In this study we extend the Hadamard's type inequalities for convex functions defined on the minimum modulus of integral functions in complex field. Firstly, using the Principal of minimum modulus theorem we derive that m (r) is continuous and decreasing function in R + . Secondly, we introduce a function t (r) and derived that t (r) and lnt (r) are continuous and convex in 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
Hadamard’s inequalities deal with a convex function f (x) on [a, b] ∈ R is between the values of f at the midpoint x = (a+b)/2 and the average of the values of f at the endpoints a and b (Chen, 2012)
The main principle of this study is to establish some integral inequality involving the modulus of complex integral functions
Summary
Let f: I ⊆ R → R is a convex mapping defined on the interval I ∈ R. Definition 2: If AB and BC are two rectifiable arc of lengths l and l', respectively, which have only the point B in common, the arc AC is rectifiable, its length being l + l' From this it follows that a Jordan arc which consists of a finite number of regular arcs is rectifiable, its length being the sum of the lengths of the regular arcs forming it. Definition 3: The maximum and minimum modulus of an integral function usually denoted by M (r) and m (r) respectively and defined by:. Theorem 1: (The Principal of Maximum Modulus Theorem) Let f (z) is analytic function, regular in a region D and on its boundary C, where C is a simple closed contour.
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