22,150 research outputs found
A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications
A class of functions involving the divided differences of the psi function
and the polygamma functions and originating from Kershaw's double inequality
are proved to be completely monotonic. As applications of these results, the
monotonicity and convexity of a function involving ratio of two gamma functions
and originating from establishment of the best upper and lower bounds in
Kershaw's double inequality are derived, two sharp double inequalities
involving ratios of double factorials are recovered, the probability integral
or error function is estimated, a double inequality for ratio of the volumes of
the unit balls in and respectively is
deduced, and a symmetrical upper and lower bounds for the gamma function in
terms of the psi function is generalized.Comment: 11 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
Some properties of extended remainder of Binet's first formula for logarithm of gamma function
In the paper, we extend Binet's first formula for the logarithm of the gamma
function and investigate some properties, including inequalities, star-shaped
and sub-additive properties and the complete monotonicity, of the extended
remainder of Binet's first formula for the logarithm of the gamma function and
related functions.Comment: 8 page
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