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

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...

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Published inMathematical and computer modelling Vol. 55; no. 7-8; pp. 1849 - 1857
Main Authors Pečarić, J., Perić, I., Roqia, G.
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.04.2012
Subjects
3-convex function
[formula omitted]-convex function
computer techniques
decision making
Exponential convexity
mathematical models
3-convex function
Exponential convexity
log-convex function
Online AccessGet full text
ISSN0895-7177
1872-9479
DOI10.1016/j.mcm.2011.11.032

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Summary: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.
Bibliography:http://dx.doi.org/10.1016/j.mcm.2011.11.032
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ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2011.11.032