LOGARITHMIC CONVEXITY OF THE ONE-PARAMETER MEAN VALUES
نویسندگان
چکیده
منابع مشابه
Research Article Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables
and let dμ= du1, . . . ,dun−1 be the differential of the volume in En−1. The weighted arithmetic mean A(x,u) and the power mean Mr(x,u) of order r with respect to the numbers x1,x2, . . . ,xn and the positive weights u1,u2, . . . ,un with ∑n i=1ui = 1 are defined, respectively, as A(x,u) = ∑ni=1uixi, Mr(x,u) = (∑ni=1uixr i ) for r =0, and M0(x,u)= ∏n i=1x ui i . For u=(1/n,1/n, . . . ,1/n), we ...
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For fixed s 1 and t1,t2 ∈ (0,1/2) we prove that the inequalities G(t1a + (1− t1)b,t1b+(1− t1)a)A1−s(a,b) > AG(a,b) and G(t2a+(1− t2)b,t2b+(1− t2)a)A1−s(a,b) > L(a,b) hold for all a,b > 0 with a = b if and only if t1 1/2− √ 2s/(4s) and t2 1/2− √ 6s/(6s) . Here G(a,b) , L(a,b) , A(a,b) and AG(a,b) are the geometric, logarithmic, arithmetic and arithmetic-geometric means of a and b , respectively....
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where x and y are positive variables and r and s are real variables. They are also called sum mean values. There has been a lot of literature such as [3, 4, 5, 6, 9, 10, 11, 12, 13, 19, 20, 21] and the related references therein about inequalities and properties of Gini means. The aim of this paper is to prove the monotonicity and logarithmic convexity of Gini means G(r, s;x, y) and related fun...
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ژورنال
عنوان ژورنال: Taiwanese Journal of Mathematics
سال: 2007
ISSN: 1027-5487
DOI: 10.11650/twjm/1500404648