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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.
Low back pain is the most common pain syndrome and a global health burden. The etiology in most cases is multifactorial and the facet joints can be a source of low back pain. The facet joint is innervated by the medial branch of the dorsal ramus of the spinal nerve. Facet joint disturbances can be responsible for 10% to 50% of all cases of chronic lumbar… (More)
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
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 .
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)
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.