On some inequalities for Beta and Gamma functions via some classical inequalities
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e improve several results recently established by Dragomir et al. in (2000) for the Gamma and Beta functions. All we need is some clever applications of classical inequalities. 1. Introduction Recently, in the survey paper [6] various inequalities for Beta and Gamma functions obtained from some classical inequalities are given. The most common way in which the Gamma function is defined is the following integral representation: Γ(x) =
∞ 0
e−t t x−1 dt,
x > 0.
(1.1)
The integral in (1.1) is uniformly convergent for all a ≤ x ≤ b, where 0 < a ≤ b < ∞, so we also have ∞
Γ(k) (x) =
0
e−t t x−1 (logt)k dt,
x > 0.
(1.2)
Various well-known formulas for Gamma function are also given in [6]. For example, Γ(x) = sx
∞ 0
e−st t x−1 dt,
x,s > 0.
(1.3)
x > 0, y > 0,
(1.4)
The Beta function is given by B(x, y) =
1 0
t x−1 (1 − t) y−1 dt,
and its connection to Gamma function is also well known: B(x, y) =
Γ(x)Γ(y) . Γ(x + y)
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 593–613 DOI: 10.1155/JIA.2005.593
(1.5)
594
Inequalities for Beta and Gamma functions
Among others know formulas for Beta function given in [6] is the following one: B(x + 1, y) + B(x, y + 1) = B(x, y),
x, y > 0.
(1.6)
Let us note that (1.6) is a special case of the following formula: n n k=0
k
B(x + k, y + n − k) = B(x, y),
x, y > 0.
(1.7)
Indeed, we have n n k=0
k
B(x + k, y + n − k) =
n 1 n k=0
= = =
k
0
t x+k−1 (1 − t) y+n−k−1 dt
1 n
n x+k−1 t (1 − t) y+n−k−1 dt k
0 k=1
1 0
1 0
t
x−1
(1 − t)
y −1
n n k=0
k
(1.8)
k
t (1 − t)
n−k
dt
t x−1 (1 − t) y−1 dt = B(x, y).
For example, the following inequalities are obtained in [6]. If p, q > 1, x ∈ [0,1], then B(p, q) − x p−1 (1 − x)q−1
(p − 2) p−2 (q − 2)q−2 1 1 ≤ max{ p − 1, q − 1} + x− (p + q − 4) p+q−4 4 2 B(p, q) − x p−1 (1 − x)q−1 ≤ max{ p − 1, q − 1}B(p − 1, q − 1)
1 1 + x− 4 2
2
.
2
,
(1.9)
(1.10)
If s > 1, p, q > 2 − 1/s > 1, 1/s + 1/r = 1, then B(p, q) − x p−1 (1 − x)q−1 ≤
r+1 1/r
1/s 1 x + (1 − x)r+1 max{ p − 1, q − 1} B s(p − 2) + 1, s(q − 2) + 1 . 1/r (r + 1) (1.11)
In this paper, we will give some improvements and generalizations of these and some other results from [6]. 2. Inequalities via Chebyshev’s inequality The following result is well known in the literature as Chebyshev’s integral inequality for synchronous (asynchronous) mappings (see, e.g., [16, pages 239–293] or [17, pages 197–208]).
R. P. Agarwal et al. 595 Lemma 2.1. Let f ,g,h : I ⊂ R → R be such that h(x) ≥ 0 for x ∈ I and h, h f g, h f and hg are integrable on I. If f and g are synchronous (asynchronous) on I, that is, if it holds
f (x) − f (y) g(x) − g(y) ≥ (≤)0 ∀x, y ∈ I,
(2.1)
then we have the inequality
I
h(x)dx
I
h(x) f (x)g(x)dx ≥ (≤)
I
h(x) f (x)dx
I
h(x)g(x)dx.
(2.2)
Theorem 2.2. Let m, p, and k be real numbers with m, p > 0 and p > k > −m, and let n be a nonnegative integer, k(p − m − k) ≥ (≤
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