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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.

MSC:
26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
26E60 Means
33B15 Gamma, beta and polygamma functions
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References:
[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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