Exponentially convex functions generated by Wulbert’s inequality and Stolarsky-type means

Abstract

Let −∞<a<b<∞. If f is concave on [a,b] and ψ′ is convex on the interval of integration, then Wulbert proved that 1δ+−δ−∫δ−δ+ψ(u)du≥1b−a∫abψ(f(x))dx, where δ−=f̄−3(‖f‖22−(f̄)2)1/2, δ+=f̄+3(‖f‖22−(f̄)2)1/2, f̄=1b−a∫abf(x)dx and ‖f‖p=(1b−a∫ab|f(x)|pdx)1/p. We define new Cauchy type means using a functional defined via above inequality and give some related results as applications.