Geodesic
Inequalities of Hermite-Hadamard type for $K$-bounded modulus convex complex functions
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2020), pp. 11-23.

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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. 
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     author = {Silvestru Sever Dragomir},
     title = {Inequalities of {Hermite-Hadamard} type for $K$-bounded modulus convex complex functions},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
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Silvestru Sever Dragomir. Inequalities of Hermite-Hadamard type for $K$-bounded modulus convex complex functions. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2020), pp. 11-23. http://geodesic.mathdoc.fr/item/BASM_2020_2_a1/

[1] Dragomir S. S., “Integral inequalities of Hermite-Hadamard type for $K$-bounded norm convex mappings”, Ukrainian Mathematical Journal, 68:10 (2017), 1530–1551 | DOI | MR | Zbl

[2] Dragomir S. S., “Ostrowski type inequalities for Lebesgue integral: a survey of recent results”, Australian J. Math. Anal. Appl., 14:1 (2017), 1, 1–287 http://ajmaa.org/cgi-bin/paper.pl?string=v14n1/V14I1P1.tex | MR | Zbl

[3] Dragomir S. S., Pearce C. E. M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, 2000 http://rgmia.org/monographs/hermite_hadamard.html