Remarks on some completely monotonic functions

Applying the Euler-Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function , x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function Γ(x), the psi function ψ(x) and the polygamma functions ψ(n)(x), n=1,2,… .     

Some completely monotonic functions involving polygamma functions and an application

By using the first Binet's formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained. As direct consequences, two inequalities are proved. As an application, the best lower and upper bounds of the nth harmonic number are established.     

Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function f admits the double Jordan-type inequality if and only if f is analytic and even. Associated with f is the function g with f(x)=g(x²). In this short note, based on this association, and using properties of absolutely and/or (completely) monotonic functions, we propose a concise method to derive the inequality from the coefficients in the Taylor’s series of f. The results include some existing ones as special cases.     

Monotonicity of some functions involving the gamma and psi functions

Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.

     

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

We introduce completely monotonic functions of order r>0 and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.     

A new upper bound in the second Kershaw's double inequality and its generalizations

In the paper, a new upper bound in the second Kershaw's double inequality involving ratio of gamma functions is established, and, as generalizations of the second Kershaw's double inequality, the divided differences of the psi and polygamma functions are bounded.     

On some properties of digamma and polygamma functions

In this note we present some new and structural inequalities for digamma, polygamma and inverse polygamma functions. We also extend, generalize and refine some known inequalities for these important functions.     

Completely monotonic functions involving the gamma and -gamma functions

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and -gamma functions.

     

A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors concisely survey and review some functionsinvolving the gamma function and its various ratios, simply state theirlogarithmically complete monotonicity and related results, and find necessaryand sufficient conditions for a new function involving the ratio of twogamma functions and originating from the coding gain to be logarithmicallycompletely monotonic.     

On Ruijsenaars' asymptotic expansion of the logarithm of the double gamma function

We study the remainder RN(x) in an asymptotic expansion due to S.N.M. Ruijsenaars, for the logarithm of the double gamma function. We show that for any even number N the function is completely monotonic on (0,∞). This proves a recent conjecture of H.L. Pedersen. In addition, we give an estimate for the remainder RN(x) for N even.     

On some properties of the gamma function

In this paper we prove a complete monotonicity theorem and establish some upper and lower bounds for the gamma function in terms of digamma and polygamma functions.     

The moment function for the ratio of correlated generalized gamma variables

The moment function for the ratio of correlated generalized gamma variables is expressed in terms of special functions. The expression presented generalizes the known moment expression for the integer valued moments to the real valued moments. Approximate formulas, in terms of elementary functions, are provided for low and high correlation regions and some application examples are given.     

Logarithmically completely monotonic functions relating to the gamma function

In this paper, the logarithmically complete monotonicity of the function exΓ(x+β)/xx+β−α in (0,∞) for α∈R and β?0 is considered and the corresponding result by G.D. Anderson, R.W. Barnard, K.C. Richards, M.K. Vamanamurthy and M. Vuorinen is generalized. As applications of these results, some inequalities between identric mean and ratio of two gamma functions by J.D. Ke?ki? and P.M. Vasi? are extended.     

Rational series for multiple zeta and log gamma functions

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We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers.

Video

For a video summary of this paper, please click here or visit http://youtu.be/2i5PQiueW_8.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

Fourier transform and distributional representation of the generalized gamma function with some applications

We present a Fourier transform representation of the generalized gamma functions, which leads to a distributional representation for them as a series of Dirac-delta functions. Applications of these representations are shown in evaluation of the integrals of products of the generalized gamma function with other functions. The results for Euler’s gamma function are deduced as special cases. The relation of the generalized gamma function with the Macdonald function is exploited to deduce new identities for it.     

Two new asymptotic expansions of the ratio of two gamma functions

Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma functions are obtained.     

Hermite-Hadamard’s type inequalities for operator convex functions

Some Hermite-Hadamard’s type inequalities for operator convex functions of selfadjoint operators in Hilbert spaces are given. Applications for particular cases of interest are also provided.     

Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as $x\to \infty$,$\frac{\mathrm{\Gamma }(x+p)\mathrm{\Gamma }(x+q)}{\mathrm{\Gamma }{(x+(p+q)/2)}^2}=\exp ?[\sum _{k=1}^n\frac{B_{2k}\left(s\right)?B_{2k}(1/2)}{k(2k?1){(x+r_0)}^{2k?1}}+R_n(x;p,q)]$where $p,q\in \mathbb{R}$ with $w=|p?q|\ne 0$ and $s=(1?w)/2$, $r_0=(p+q?1)/2$, $B_{2n+1}\left(s\right)$ are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function $x?{(?1)}^n{R}_n(x;p,q)$ is completely monotonic on $(?r_0,\infty )$ if $|p?q|<1$, which yields a sharp upper bound for $\left|R_n(x;p,q)\right|$. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when $q\to p$.