The Hermite-Hadamard inequality is used to develop an approximation to the logarithm of the gamma function which is more accurate than the Stirling approximation and easier to derive. Then the concavity of the logarithm of gamma of logarithm is proved and applied to the Jensen inequality. Finally, the Wallis ratio is used to obtain the additional term in… (More)
Asymptotic expansions of the multiple quotients of two gamma functions are obtained and analyzed. We apply these results to the hypergeometric function and central multinomial coefficient which leads to the new approximation formulas.
The subject of this paper is a systematic study of inequalities of the form which cover Neuman-Sándor mean and some classical means. Furthermore , following
Let s , t be two given real numbers , s = t and m ∈ N. We determine the coefficients a j (s, t) in the asymptotic expansion of integral (or differential) mean of polygamma functions ψ (m) (x) : 1 t − s t s ψ (m) (x + u) d u ∼ ψ (m) x ∞ ∑ j=0 a j (s, t) x j , x → ∞. We derive the recursive relations for polynomials a j (t , s) , and also as polynomials in… (More)
Asymptotic expansion of the arithmetic-geometric mean is obtained and it is used to analyze inequalities and relations between arithmetic-geometric mean and other classical means .
We give a systematic view of the asymptotic expansion of two well-known sequences, the central binomial coefficients and the Catalan numbers. The main point is explanation of the nature of the best shift in variable n, in order to obtain " nice " asymptotic expansions. We also give a complete asymptotic expansion of partial sums of these sequences.
We give an overview of the use of asymptotic expansions of gamma and related functions — ratio of gamma functions, powers, digamma and polygamma functions. The aim is to introduce a general theory which can unify various particular formulas for factorial functions and binomial coefficients. The connection with inequalities for gamma function is established.… (More)