Results 81 to 90 of about 5,395,506 (201)

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Journal of Inequalities and Applications, 2019
Let Hn=∑r=1n1/r $H_{n} = \sum_{r=1}^{n} 1/r$ and Hn(x)=∑r=1n1/(r+x) $H_{n}(x) = \sum_{r=1}^{n} 1/(r+x)$. Let ψ(x) $\psi(x)$ denote the digamma function. It is shown that Hn(x)+ψ(x+1) $H_{n}(x) + \psi(x+1)$ is approximated by 12logf(n+x) $\frac{1}{2}\log ...
G. J. O. Jameson
doaj   +1 more source

A completely monotonic function involving the gamma and trigamma functions

Kuwait Journal of Science, 2016
In the paper the author provides necessary and sufficient conditions on $a$ for the function\begin{equation*}\frac{1}{2}\ln(2\pi)-x+\biggl(x-\frac{1}{2}\biggr)\ln x-\ln\Gamma(x)+\frac1{12}{\psi'(x+a)}\end{equation*}and its negative to be completely ...
Feng Qi
doaj  

Inequalities for ζ(s) − ψ(1 − s) Related to a Conjecture of Henry

Axioms
In this paper, we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function.
Liwen Gao
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On complete monotonicity for several classes of functions related to ratios of gamma functions

Journal of Inequalities and Applications, 2019
Let Γ(x) $\varGamma (x)$ denote the classical Euler gamma function. The logarithmic derivative ψ(x)=[lnΓ(x)]′=Γ′(x)Γ(x) $\psi (x)=[\ln \varGamma (x)]'=\frac{\varGamma '(x)}{ \varGamma (x)}$, ψ′(x) $\psi '(x)$, and ψ″(x) $\psi ''(x)$ are, respectively ...
Feng Qi, Ravi P. Agarwal
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Integral and series representations of the digamma and polygamma functions [PDF]

Analysis, 2012
We obtain a variety of series and integral representations of the digamma function $ψ(a)$. These in turn provide representations of the evaluations $ψ(p/q)$ at rational argument and for the polygamma function $ψ^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $γ_0(a)=-ψ(a)$.
openaire   +2 more sources

Analysis On Simpson's Type Inequalities Through Generalized Convexity with Applications [PDF]

Sahand Communications in Mathematical Analysis
Fractional operators and integral inequalities have become a focal point of research due to their applications in mathematics, physics, engineering, and applied sciences.
Arslan Munir, Artion Kashuri
doaj   +1 more source

New Inequalities for Gamma and Digamma Functions [PDF]

, 2020
By using the mean value theorem and logarithmic convexity, we obtain some new inequalities for gamma and digamma ...
M R Farhangdoost, M Kargar Dolatabadi
core  

A study of Hermite-Hadamard inequalities via Caputo-Fabrizio fractional integral operators using strongly ( s , m ) $(s, m)$ -convex functions in the second sense

Journal of Inequalities and Applications
New ways for comparing and bounding strongly ( s , m ) $(s,m)$ -convex functions using Caputo fractional derivatives and Caputo-Fabrizio integral operators are explored.
Jie Li, Yong Lin, Serap Özcan, Muhammad Shoaib Saleem, Ahsan Fareed Shah   +4 more
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Correction: On the positivity of a certain function related with the Digamma function

The Ramanujan Journal
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Geometric Properties of the Normalized Fractional Bessel-Maitland Function Associated with the Riemann–Liouville Fractional Derivative

Fractal and Fractional
In this paper, we introduce and investigate a normalized fractional Bessel–Maitland function generated through the Riemann–Liouville fractional derivative, thereby establishing a new connection between fractional calculus and geometric function theory ...
Javeria, Alina Alb Lupaş, Nazar Khan
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