22,120 research outputs found
A new way to prove L'Hospital Monotone Rules with applications
Let . Let and be differentiable
functions on and let on . By introducing an
auxiliary function , 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 and convexity of a function involving digamma one and their applications
Let be defined on or 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 , where denotes the Psi function. And, we
determine the best parameter such that the inequality \psi \left(
x+1\right) \right) \mathcal{L}% (x,a) holds for or , and then, some new and very
high accurate sharp bounds for pis function and harmonic numbers are presented.
As applications, we construct a sequence
defined by , which
gives extremely accurate values for .Comment: 20 page
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