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Zhen-Hang Yang

Publications and source records attributed to Zhen-Hang Yang.

29 records · Page 2Linked to original sources
arXiv2014

On Lazarevic and Cusa type inequalities for hyperbolic functions with two parameters

In this paper, by investigating the monotonicity of a function composed of $% \left( \sinh x\right) /x$ and $\cosh x$ with two parameters in $x$ on $% \left( 0,\infty \right) $, we prove serval theorems related to inequalities for hyperbolic functions, which generalize known results and establish some new and sharp inequalities. As applications, some new and sharp inequalities for bivariate means are presented.

math.CA↗
arXiv2014

Sharp Cusa type inequalities for trigonometric functions with two parameters

Let $\left( p,q\right) \mapsto β\left( p,q\right) $ be a function defined on $\mathbb{R}^{2}$. We determine the best or better $p,q$ such that the inequality% \begin{equation*} \left( \frac{\sin x}{x}\right) ^{p}<\left( >\right) 1-β\left( p,q\right) +β\left( p,q\right) \cos ^{q}x \end{equation*}% holds for $x\in \left( 0,π/2\right) $, and obtain a lot of new and sharp Cusa type inequalities for trigonometric functions. As applications, some new Shafer-Fink type and Carlson type inequalities for arc sine and arc cosine functions, and new inequalities for trigonometric means are established.

math.CA↗
arXiv2014

Bivariate log-convexity of the more extended means and its applications

In this paper, the bivariate log-convexity of the two-parameter homogeneous function in parameter pair is vestigated. From this the bivariate log-convexity of the more extended means with respect to a parameter pair is solved. It follows that Stolarsky means, Gini means, two-parameter identric (exponential) means and two-parameter Heronian means are all bivariate log-concave on $\mathbb{[}0\mathbb{,\infty )}^{2}$ and log-convex on $% \mathbb{(-\infty ,}0\mathbb{]}^{2}$ with respect to parameters. Lastly, some classical and new inequalities for means are given.

math.CA↗
arXiv2013

Some sharp Wilker type inequalities and their applications

In this paper, we prove that for fixed $k\geq 1$, the Wilker type inequality {equation*} \frac{2}{k+2}(\frac{\sin x}{x}) ^{kp}+\frac{k}{k+2}(\frac{% \tan x}{x})^{p}>1 {equation*}% holds for $x\in (0,π/2) $ if and only if $p>0$ or $p\leq -% \frac{\ln (k+2) -\ln 2}{k(\ln π-\ln 2)}$. It is reversed if and only if $-\frac{12}{5(k+2)}\leq p<0$. Its hyperbolic version holds for $x\in (0,\infty) $ if and only if $% p>0$ or $p\leq -\frac{12}{5(k+2)}$. And, for fixed $k<-2$, the hyperbolic version is reversed if and only if $p<0$ or $p\geq -\frac{12}{% 5(k+2)}$. Our results unify and generalize some known ones.

math.CA↗
arXiv2013

Unification and refinements of Jordan, Adamović-Mitrinovićand and Cusa's inequalities

In this paper, we find some new sharp bounds for $\left(\sin x\right) /x$, which unify and refine Jordan, Adamović-Mitrinovićand and Cusa's inequalities. As applications of main results, some new Shafer-Fink type inequalities for arc sine function and ones for certain bivariate means are established, and a simpler but more accurate estimate for sine integral is derived.

math.CA↗
arXiv2012

The monotonicity results and sharp inequalities for some power-type means of two arguments

For $a,b>0$ with $a\neq b$, we define M_{p}=M^{1/p}(a^{p},b^{p})\text{if}p\neq 0 \text{and} M_{0}=\sqrt{ab}, where $M=A,He,L,I,P,T,N,Z$ and $Y$ stand for the arithmetic mean, Heronian mean, logarithmic mean, identric (exponential) mean, the first Seiffert mean, the second Seiffert mean, Neuman-Sándor mean, power-exponential mean and exponential-geometric mean, respectively. Generally, if $M$ is a mean of $a$ and $b$, then $M_{p}$ is also, and call "power-type mean". We prove the power-type means $P_{p}$, $T_{p}$, $N_{p}$, $Z_{p}$ are increasing in $p$ on $\mathbb{R}$ and establish sharp inequalities among power-type means $A_{p}$, $He_{p}$, $L_{p}$, $I_{p}$, $P_{p}$, $N_{p}$, $Z_{p}$, $Y_{p}$% . From this a very nice chain of inequalities for these means L_{2}<P<N_{1/2}<He<A_{2/3}<I<Z_{1/3}<Y_{1/2} follows. Lastly, a conjecture is proposed.

math.CA↗
arXiv2012

Sharp power means bounds for Neuman-Sándor mean

For a,b>0 with a\not=b, let N(a,b) denote the Neuman-Sándor mean defined by N(a,b)=((a-b)/(2arcsinh((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp power means bounds for the Neuman-Sándor mean: A_{p_1}(a,b)<N(a,b)\leqA_{p_2}(a,b), where p_1= ((ln2)/(lnln(3+2\surd2))) and p_2=4/3 are the best constants.

math.CA↗
arXiv2012

Refinements of Mitrinović-Cusa inequality

The Mitrinović-Cusa inequality states that for x\in(0,π/2) (cos x)^{1/3}<((sin x)/x)<((2+cos x)/3) hold. In this paper, we prove that (cos x)^{1/3}<(cos px)^{1/(3p^{2})}<((sin x)/x)<(cos qx)^{1/(3q^{2})}<((2+cos x)/3) hold for x\in(0,π/2) if and only if p\in[p_{1},1) and q\in(0,1/\surd5], where p_{1}=0.45346830977067.... And the function p\mapsto(cos px)^{1/(3p^{2})} is decreasing on (0,1]. Our results greatly refine the Mitrinović-Cusa inequality.

math.CA↗
arXiv2012

Sharp bounds for the second Seiffert mean in terms of power means

For a,b>0 with a\not=b, let T(a,b) denote the second Seiffert mean defined by T(a,b)=((a-b)/(2arctan((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp bounds for the second Seiffert mean in terms of power means: A_{p_1}(a,b)<T(a,b)\leqA_{p_2}(a,b), where p_1= log_{π/2}2 and p_2=5/3 can not be improved.

math.CA↗
arXiv2012

New sharp Jordan type inequalities and their applications

In this paper, we prove that for x\in(0,π/2) (cos p_0x)^{1/p_0}<((sin x)/x)<(cos(x/3))^3 with the best constants p_0=0.347307245464... and 1/3. Moreover, if p\in (0,1/3] then the double inequality β_{p}(cos px)^{1/p}<((sin x)/x)<(cos px)^{1/p} holds for x\in(0,π/2), where β_{p}=2π^-1(cos((pπ)/2))^{-1/p} and 1 are the best possible. Its reverse one holds if p\in[1/2,1]. As applications, some new inequalities are established.

math.CA↗