Generalized convexity and inequalities
Copyright policy )Generalized convexity and inequalities
Let R+ = (0,infinity) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 in M, we say that a function f : R+ to R+ is (m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in R+ . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient conditions for (m1,m2)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.
17 pages
- University of Turku Finland
- University of Auckland New Zealand
- Michigan State University United States
Monotonicity, Approximation to limiting values (summation of series, etc.), Generalized hypergeometric series, Convexity of real functions in one variable, generalizations, Convexity, Classical hypergeometric functions, \({}_2F_1\), Euler-Maclaurin formula in numerical analysis, 33C20 (Primary), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Hypergeometric function, 33C05; 33C20 (Primary); 26A51 (Secondary), Generalized convexity, Applied Mathematics, ta111, Power series, Convergence and divergence of series and sequences of functions, 33C05, 26A51 (Secondary), Maclaurin series, Mathematics - Classical Analysis and ODEs, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Analysis, hypergeometric functions
Monotonicity, Approximation to limiting values (summation of series, etc.), Generalized hypergeometric series, Convexity of real functions in one variable, generalizations, Convexity, Classical hypergeometric functions, \({}_2F_1\), Euler-Maclaurin formula in numerical analysis, 33C20 (Primary), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Hypergeometric function, 33C05; 33C20 (Primary); 26A51 (Secondary), Generalized convexity, Applied Mathematics, ta111, Power series, Convergence and divergence of series and sequences of functions, 33C05, 26A51 (Secondary), Maclaurin series, Mathematics - Classical Analysis and ODEs, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Analysis, hypergeometric functions
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).125 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 1% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 1% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10%
Citations provided by BIP!Popularity provided by BIP!