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ЖУРНАЛЫ // Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica // Архив

Bul. Acad. Ştiinţe Repub. Mold. Mat., 2020, номер 2, страницы 11–23 (Mi basm529)

Research articles

Inequalities of Hermite-Hadamard type for $K$-bounded modulus convex complex functions

Silvestru Sever Dragomir

College of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia

Аннотация: Let $D\subset \mathbb{C}$ be a convex domain of complex numbers and $K>0.$ We say that the function $f:D\subset \mathbb{C\rightarrow C}$ is called $K$-bounded modulus convex, for the given $K>0,$ if it satisfies the condition
\begin{equation*} \left\vert \left( 1-\lambda \right) f\left( x\right) +\lambda f\left( y\right) -f\left( \left( 1-\lambda \right) x+\lambda y\right) \right\vert \leq \frac{1}{2}K\lambda \left( 1-\lambda \right) \left\vert x-y\right\vert ^{2} \end{equation*}
for any $x,$ $y\in D$ and $\lambda \in \left[ 0,1\right] .$ In this paper we establish some new Hermite-Hadamard type inequalities for the complex integral on $\gamma ,$ a smooth path from $\mathbb{C}$, and $K$-bounded modulus convex functions. Some examples for integrals on segments and circular paths are also given.

Ключевые слова и фразы: complex integral, continuous functions, holomorphic functions, hermite-Hadamard inequality, midpoint inequality, trapezoid inequality.

MSC: 26D15, 26D10, 30A10, 30A86

Поступила в редакцию: 11.09.2019

Язык публикации: английский



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