$$P$$-functions, quasi-convex functions, and Hadamard-type inequalities.(English)Zbl 0939.26009

A nonnegative function $$f$$ defined on $$[a,b]$$ is said to be a $$P$$-function if $f(tx+ (1- t)y)\leq f(x)+ f(y),\quad x,y\in [a,b],\quad t\in [0,1].$ The authors show that the set of all $$P$$-functions on $$[a,b]$$ is the least set closed under pointwise sum, supremum, and convergence and containing the class of all nonnegative quasi-convex functions. Hadamard type inequalities for $$P$$-functions are also presented.

MSC:

 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations
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References:

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