\(P\)-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.


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