Abstract
In the expository review and survey paper dealing with bounds for the ratio of two gamma functions, along one of the main lines of bounding the ratio of two gamma functions, the authors look back and analyze some known results, including Wendel’s asymptotic relation, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Kershaw’s, and Elezović-Giordano-Pečarić’s inequalities, Lazarević-Lupaş’s claim, and other monotonic and convex properties. On the other hand, the authors introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years.MSC: 33B15, 26A48, 26A51, 26D07, 26D15, 44A10.
Highlights
Recall [, Chapter XIII] and [, Chapter IV] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (– )nf (n)(x) ≥ ( )for x ∈ I and n ≥
It is well known that the classical Euler gamma function may be defined for x > by
(x + m + ) ≤ (x + s) –s (x + n + s) mi= (x + i) ni=– (x + i + s) for x > and < s
Summary
Recall [ , Chapter XIII] and [ , Chapter IV] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (– )nf (n)(x) ≥. (x + m + ) ≤ (x + s) –s (x + n + s) mi= (x + i) ni=– (x + i + s) for x > and < s < , where m and n are positive integers This implies that basing on recurrence formula ( ) and double inequality ( ), one can bound the ratio (x+a) (x+b) for any positive numbers x, a and b. Remark There has been a lot of literature about asymptotic expansions of a ratio of two gamma functions Because this expository review and survey paper is devoted to inequalities and complete monotonicity, we will not further survey the topic of asymptotic expansions of a quotient of two gamma functions.
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