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A note on Schur-convex functions. (English) Zbl 0978.26013
Let \(f\) be a continuous function on an interval \(I\), and \[ F(x,y)= {1\over y-x} \int^y_x f(t) dt,\quad x,y\in I,\quad x\neq y,\quad F(x,x)= f(x). \] The authors prove that \(F\) is Schur-convex if and only if \(f\) is convex. Applications to Schur-convexity of the logarithmic mean and the gamma function are given.

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
26E60 Means
33B15 Gamma, beta and polygamma functions
Full Text: DOI Link
[1] M. Abramowitz and I.A. Stegun (eds.), Handbook of mathematical functions , 4th printing, National Bureau of Standards, Washington, DC, 1965.
[2] A.W. Marshall and I. Olkin, Inequalities : Theory of majorization and its applications , Academic Press, New York, 1979. · Zbl 0437.26007
[3] M. Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function , Rocky Mountain J. Math. 28 (1998), 1053-1066. · Zbl 0928.26012
[4] A.W. Roberts and D.E. Varberg, Convex functions , Academic Press, New York, 1973. · Zbl 0271.26009
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