Log-convexity and log-concavity for series in gamma ratios and applications

Abstract

Polynomial sequence {P m } m≥0 is q-logarithmically concave if P 2 m − P m+1 P m−1 is a polynomial with nonnegative coefficients for any m ≥ 1. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of q-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.

Analytic Number Theory, Approximation Theory, and Special Functions

“…A little later Deshouillers and Iwaniec [17] studied, in the above notation, which is incidentally equivalent to the Lindelöf hypothesis (if log N log T ). An improvement of (165) was obtained by Watt [104] who proved the bound I.T; N / " T " .1 C N 2 T 1=2 / max n6N ja n j 2 :…”

“…The relatively short range for N , namely N 6 T 1=11 " , is compensated by the fact that one indeed obtains an asymptotic formula and not just an upper bound as was done in previous works. For example, Watt [104] obtains the desired upper bound for N T 1=4 , but his method does not produce an asymptotic formula for the integral K.T; N /.…”

“…x log 4 T " T 1=2C" ; (104) and even the weak F .T / T 2 log C T (implied by Theorem 4.4) gives the bound in Theorem 4.3. One also obtains from F .T / D O " .T 3=2C" / the (hitherto unproved) bound…”

“…The sequence f k g is known to satisfy the condition as in (66), but the product in (68) is always divergent; so, in order for the expression (68) to make sense, some sort of regularization procedure must be used (see e.g., [63,104,116,120]). It is easily seen that, formally, e Z 0 .0/ is the product of nonzero eigenvalues of .…”

Efficient approximations of the gamma function and further properties

“…Only in the recent past, many formulas were presented. We refer for example to Batir and Chen (2012), Batir (2010), Burnside (1917), Chen (2013), Chen and Lin (2012), Chen and Mortici (2012), Dubourdieu (1939), Gosper (1978), Communicated Kalmykov and Karp (2013), Laforgia and Natalini (2013), , , Lu and Wang (2013), Mortici (2009), Mortici (2010), Nemes (2012), where also estimates for polygamma and other related functions were stated. Starting from the Stirling's formula (x + 1) ∼ √ 2π x x+ 1 2 e −x (1.1) and Burnside's formula (Burnside 1917)…”