Abstract
AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$
Γ
q
(
x
+
1
)
/
Γ
q
(
x
+
s
)
for all real number s and $0< q\neq1$
0
<
q
≠
1
in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$
0
<
s
<
1
when letting $q\to 1$
q
→
1
and a new inequality for all $s>1$
s
>
1
.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference36 articles.
1. Ismail, M.E.H., Muldoon, M.E.: Inequalities and monotonicity properties for gamma and q-gamma functions, approximation and computation. In: Internat. Ser. Numer. Math., vol. 119, pp. 309–323. Birkhäuser, Boston (1994)
2. Alzer, H.: Sharp bounds for the ratio of q-gamma functions. Math. Nachr. 222(1), 5–14 (2001)
3. Mortici, C.: New approximation formulas for evaluating the ratio of gamma functions. Math. Comput. Model. 52, 425–433 (2010)
4. Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)
5. Kershaw, D.: Some extensions of W Gautschi’s inequalities for the gamma function. Math. Comput. 41, 607–611 (1983)
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