arXiv subjects
Publications and source records attributed to Yu-Ming Chu.
In this paper, we prove that the double inequality $M_{p}(a,b) 0$ with $a\neq b$ if and only if $p\leq 4\log 2/(4+2\log 2-π)=1.2351\cdots$ and $q\geq 4/3$, where $% M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ $(r\neq 0)$ and $M_{0}(a,b)=\sqrt{ab}$ is the $r$th power mean, $B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ is the Sá% ndor-Yang mean, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ and $% T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$.
In this paper, we find the greatest values $α_{1}$, $α_{2}$, $α_{3}$, $α_{4}$, $α_{5}$, $α_{6}$, $α_{7}$, $α_{8}$ and the least values $β_{1}$, $β_{2}$, $β_{3}$, $β_{4}$, $β_{5}$, $β_{6}$, $β_{7}$, $β_{8}$ such that the double inequalities $$A^{α_{1}}(a,b)G^{1-α_{1}}(a,b) 0$ with $a\neq b$, where $G$, $A$ and $Q$ are respectively the geometric, arithmetic and quadratic means, and $N_{GA}$, $N_{AG}$, $N_{AQ}$ and $N_{QA}$ are the Neuman means derived from the Schwab-Borchardt mean.
In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic mean $A$ and contraharmonic mean $C$.
We prove that the inequalities $\sin_{p,q}(\sqrt{rs})\geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ and $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ hold for all $p,q\in(1,\infty)$, $r,s\in(0,\int_{0}^{1}(1-t^q)^{-1/p}dt)$ and $r^*,s^*\in(0,\int_{0}^{\infty}(1+t^q)^{-1/p}dt)$, where $\sin_{p,q}$ and $\sinh_{p,q}$ are the generalized trigonometric and hyperbolic sine functions, respectively. As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)].
In this article, we present the best possible upper and lower bounds for the Neuman-Sándor mean in terms of the geometric combinations of harmonic and quadratic means, geometric and quadratic means, harmonic and contra-harmonic means, and geometric and contra-harmonic means.
In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.
In this article, we prove that the double inequality $$αG(a,b)+(1-α)C(a,b) 0$ with $a\neq b$ if and only if $α\geq 5/9$ and $β\leq 1-1/[2\log(1+\sqrt{2})]=0.4327...$, where $G(a,b),C(a,b)$ and $M(a,b)$ are respectively the geometric, contra-harmonic and Neuman-Sándor means of $a$ and $b$.
In this paper, we present the greatest values $α$, $λ$ and $p$, and the least values $β$, $μ$ and $q$ such that the double inequalities $αD(a,b)+(1-α)H(a,b) 0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respectively.
In this paper, some inequalities of bounds for the Neuman-Sándor mean in terms of weighted arithmetic means of two bivariate means are established. Bounds involving weighted arithmetic means are sharp.
In this paper, we find the greatest values $α$ and $λ$, and the least values $β$ and $μ$ such that the double inequalities $$C^α(a,b)A^{1-α}(a,b) 0$ with $a\neq b$, where $M(a,b)$, $A(a,b)$ and $C(a,b)$ denote the Neuman-Sándor, arithmetic, and contra-harmonic means of $a$ and $b$, respectively.
For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq 1/2-\sqrt{2s}/(4s)$ and $t_{2}\geq 1/2-\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively.
In this note we obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).
In this paper, we prove that the inequalities $α[1/3 Q(a,b)+2/3 A(a,b)]+(1-α)Q^{1/3}(a,b)A^{2/3}(a,b) 0$ with $a\neq b$ if and only if $α\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777...$, $β\geq 4/5$, $λ\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274...$, and $μ\geq 8/25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ denote the Neuman-Sándor, arithmetic, contra-harmonic, and quadratic means of $a$ and $b$, respectively.
In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>π[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all $r\in(0,1)$ proposed by M. Vuorinen.