## Hadamard-type inequalities for generalized convex functions.(English)Zbl 1043.26009

Let $$I\subset \mathbb{R}$$ be an interval and $$\omega_1,\omega_2:I\longrightarrow\mathbb{R}$$ given functions. A function $$f:I\longrightarrow\mathbb{R}$$ is called $$(\omega_1,\omega_2)$$-convex if $\begin{vmatrix} f(x)&f(y)&f(z)\\ \omega_1(x)&\omega_1(y)&\omega_1(z)\\ \omega_2(x)&\omega_2(y)&\omega_2(z)\end{vmatrix}\geq 0$ whenever $$x<y<z,\;x;y;z\in I.$$ The $$(1,x)$$-convexity is the ordinary convexity. Several characterizations of the $$(\omega_1,\omega_2)$$-convex functions are given, generalizing the known characterizations of ordinary convex functions. The paper contains also Hadamard-type inequalities for $$(\omega_1,\omega_2)$$-convex functions.

### MSC:

 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations 26B25 Convexity of real functions of several variables, generalizations
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