Sharp inequalities for Toader mean in terms of other bivariate means

Authors : Weidong JIANG

Pages : 841-849

Doi:10.15672/hujms.1106426

View : 439 | Download : 316

Publication Date : 2023-08-15

Article Type : Research Paper

Abstract :In the paper, the author discovers the best constants $\\alpha_1$, $\\alpha_2$, $\\alpha_3$, $\\beta_1$, $\\beta_2$ and $\\beta_3$ for the double inequalities \\[ \\alpha_1 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} < Tinsert ignore into journalissuearticles values(a,b);-\\frac{1}{4}C-\\frac{3}{4}A-A\\sum_{k=1}^{n-1}\\frac{insert ignore into journalissuearticles values(\\frac{1}{2},k);^2}{4insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(k+1);!);^2}\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2k+2}<\\beta_1 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} \\] \\[ \\alpha_2 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} < Tinsert ignore into journalissuearticles values(a,b);-\\frac{3}{4}\\overline{C}-\\frac{1}{4}A-A\\sum_{k=1}^{n-1}\\frac{insert ignore into journalissuearticles values(\\frac{1}{2},k);^2}{4insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(k+1);!);^2}\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2k+2}<\\beta_2 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} \\] and \\[ \\alpha_3 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} < \\frac{4}{5}Tinsert ignore into journalissuearticles values(a,b);+\\frac{1}{5}H-A-A\\sum_{k=1}^{n-1}\\frac{insert ignore into journalissuearticles values(\\frac{1}{2},k);^2}{5insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(k+1);!);^2}\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2k+2}<\\beta_3 A\\leftinsert ignore into journalissuearticles values(\\frac{a-b}{a+b}\\right);^{2n+2} \\] to be valid for all $a,b>0$ with $a\\ne b$ and $n=1,2,\\cdots$, where \\[ C\\equiv Cinsert ignore into journalissuearticles values(a,b);=\\frac{a^2+b^2}{a+b},\\,\\overline{C}\\equiv\\overline{C}insert ignore into journalissuearticles values(a,b);=\\frac{2insert ignore into journalissuearticles values(a^2+ab+b^2);}{3insert ignore into journalissuearticles values(a+b);},\\, A\\equiv Ainsert ignore into journalissuearticles values(a,b);=\\frac{a+b}{2}, \\] \\[ H\\equiv Hinsert ignore into journalissuearticles values(a,b); =\\frac{2ab}{a+b},\\quad Tinsert ignore into journalissuearticles values(a,b);=\\frac2{\\pi}\\int_0^{\\pi/2}\\sqrt{a^2\\cos^2\\theta+b^2\\sin^2\\theta}\\,{\\rm d}\\theta \\] are respectively the contraharmonic, centroidal, arithmetic, harmonic and Toader means of two positive numbers $a$ and $b$, $ insert ignore into journalissuearticles values(a,n);=ainsert ignore into journalissuearticles values(a+1);insert ignore into journalissuearticles values(a+2);insert ignore into journalissuearticles values(a+3);\\cdots insert ignore into journalissuearticles values(a+n-1);$ denotes the shifted factorial function. As an application of the above inequalities, the author also find a new bounds for the complete elliptic integral of the second kind.
Keywords : Toader mean, complete elliptic integrals, arithmetic mean, centroidal mean, contraharmonic mean