Some two-sided inequalities for multiple Gamma functions and related results
نویسندگان
چکیده
There is an abundant literature on inequalities for the (Euler’s) Gamma function C and its various related functions. Yet, only very recently, several authors began to study inequalities for the (Barnes’) double Gamma function C2. Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ. In our investigation of these two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ, we employ and extend a method based upon Taylor’s formula and express logCnð1þ xÞ as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions Cn ðn 2 NÞ;N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions Cn ðn 1⁄4 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n=6. 2013 Elsevier Inc. All rights reserved.
منابع مشابه
New inequalities for a class of differentiable functions
In this paper, we use the Riemann-Liouville fractionalintegrals to establish some new integral inequalities related toChebyshev's functional in the case of two differentiable functions.
متن کاملBounds for the Ratio of Two Gamma Functions—from Gautschi’s and Kershaw’s Inequalities to Completely Monotonic Functions
Abstract. In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two ...
متن کاملBounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity
Abstract In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio...
متن کاملThe System of Vector Variational-like Inequalities with Weakly Relaxed ${eta_gamma-alpha_gamma}_{gamma inGamma}$ Pseudomonotone Mappings in Banach Spaces
In this paper, we introduce two concepts of weakly relaxed ${eta_gamma-alpha_gamma}_{gamma in Gamma}$ pseudomonotone and demipseudomonotone mappings in Banach spaces. Then we obtain some results of the solutions existence for a system of vector variational-like inequalities with weakly relaxed ${eta_gamma-alpha_gamma}_{gamma in Gamma}$ pseudomonotone and demipseudomonotone mappings in reflexive...
متن کاملOn zero-divisor graphs of quotient rings and complemented zero-divisor graphs
For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Applied Mathematics and Computation
دوره 219 شماره
صفحات -
تاریخ انتشار 2013