501 research outputs found

### The monotonicity and convexity of a function involving digamma one and their applications

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left(
4/15,\infty \right)$ or $\left( 0,\infty \right) \times \left( 1/15,\infty
\right)$ by the formula% \begin{equation*}
\mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}%
\right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{%
15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and
convexity of the function $x\rightarrow F_{a}\left( x\right) =\psi \left(
x+1\right) -\mathcal{L}(x,a)$, where $\psi$ denotes the Psi function. And, we
determine the best parameter $a$ such that the inequality \psi \left(
x+1\right) \right) \mathcal{L}% (x,a) holds for $x\in \left(
-1,\infty \right)$ or $\left( 0,\infty \right)$, and then, some new and very
high accurate sharp bounds for pis function and harmonic numbers are presented.
As applications, we construct a sequence $\left( l_{n}\left( a\right) \right)$
defined by $l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right)$, which
gives extremely accurate values for $\gamma$.Comment: 20 page

### A new way to prove L'Hospital Monotone Rules with applications

Let $-\infty \leq a<b\leq \infty$. Let $f$ and $g$ be differentiable
functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an
auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we
easily prove L'Hoipital rules for monotonicity. This offer a natural and
concise way so that those rules are easier to be understood. Using our
L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new
sharp inequalities for hyperbolic and trigonometric functions as well as
bivariate means, which supplement certain known results.Comment: 19 page

### 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-S\'{a}ndor 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.Comment: 11 page

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