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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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