volume 6, issue 2, article 48, 2005.
Received 07 March, 2005;
accepted 19 March, 2005.
Communicated by:Th.M. Rassias
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Journal of Inequalities in Pure and Applied Mathematics
A GEOMETRICAL PROOF OF A NEW INEQUALITY FOR THE GAMMA FUNCTION
C. ALSINA AND M.S. TOMÁS
Secció de Matemàtiques
ETSAB. Univ. Politècnica Catalunya Diagonal 649, 08028 Barcelona, Spain.
EMail:claudio.alsina@upc.es EMail:maria.santos.tomas@upc.es
c
2000Victoria University ISSN (electronic): 1443-5756 085-05
A Geometrical Proof of a New Inequality for the Gamma
Function C. Alsina and M.S. Tomás
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J. Ineq. Pure and Appl. Math. 6(2) Art. 48, 2005
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Abstract
Using the inclusions between the unit balls for thep-norms, we obtain a new inequality for the gamma function.
2000 Mathematics Subject Classification:33B15, 33C05, 26D07.
Key words: Gamma function, Hypergeometric function, Inequalities.
The authors thank Prof. Hans Heinrich Kairies for his interesting remarks concerning this paper.
Since the gamma function Γ(x) =
Z ∞
0
e−ttx−1dt, x >0
is one of the most important functions in Mathematics, there exists an extensive literature on its inequalities (see [1], [2]).
Our aim here is to present and prove the inequalities 1
n! ≤ Γ(1 +x)n
Γ(1 +nx) ≤1 x∈[0,1], n∈N.
As we will show the above inequalities follow immediately from a key geo- metrical argument. From now on for any r > 0,p, n ≥ 1we will consider the notation:
Dk·kn,r
p ={(x1, . . . , xn)∈Rn/k(x1, . . . , xn)kp < r}
A Geometrical Proof of a New Inequality for the Gamma
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for then-ball of radiusrfor thep-normk(x1, . . . , xn)kp = (|x1|p+· · ·+|xn|p)1/p. To this end, we need to prove the following:
Lemma 1. For allninN,p≥1andr >0we have:
(1) Volume
Dk·kn,r
p
= 2n Γ
1 + 1pn
Γ
1 + np rn.
Proof. Forn = 1,D1,rk·k
p is the interval(−r, r), whose measure is2r, i.e., 2r= 2
Γ
1 + 1p Γ
1 + 1pr
and (1) holds. By induction, let us assume that (1) holds forn−1. Then we note that|x1|p+· · ·+|xn|p < rp is equivalent to|x1|p+· · ·+|xn−1|p < rp− |xn|p and by virtue of the induction hypothesis we have
Volume
Dn,rk·k
p
= Z
Dn,rk·k
p
dx1. . . dxn
= 2 Z r
0
Z
Dn−1,(rp−|xn|p)1/p k·kp
dx1. . . dxn−1
! dxn
A Geometrical Proof of a New Inequality for the Gamma
Function C. Alsina and M.S. Tomás
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= 2 Z r
0
2n−1 Γ
1 + 1p
n−1
Γ
1 + n−1p (rp−xpn)n−1p dxn
= 2n Γ
1 + 1pn−1
Γ
1 + n−1p rn Z 1
0
(1−zp)n−1p dz,
wherez =xn/r.
If we considerF(a, b, c, z)the first hypergeometric function (see [3]), then Z
(1−zp)n−1p dz =zF 1
p,−n−1
p ,1 + 1 p, zn
and by well-known properties of the hypergeometric function we deduce:
Volume Dk·kn,r
p
= 2n Γ
1 + 1p
n−1
Γ
1 + n−1p rnF 1
p,−n−1 p ,1 + 1
p,1
= 2n Γ
1 + 1pn−1
Γ
1 + n−1p rn Γ
1 + 1p Γ
1 + n−1p Γ
1 + np
= 2n Γ
1 + 1pn
Γ
1 + nprn.
A Geometrical Proof of a New Inequality for the Gamma
Function C. Alsina and M.S. Tomás
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Therefore we have
Theorem 2. For alln ∈Nandxin(0,1)we have
1
n! ≤ Γ(1 +x)n Γ(1 +nx) ≤1.
Proof. For allninNandp≥1, from the inclusions Dn,1k·k
1 ⊆Dk·kn,1
p ⊆Dk·kn,1
∞, we deduce
Volume
Dk·kn,1
1
≤Volume
Dk·kn,1
p
≤Volume
Dn,1k·k
∞
,
so by Lemma1:
2n Γ(2)n
Γ(n+ 1) ≤2n Γ
1 + 1p
n
Γ
1 + np ≤2n
and with1/p=x, bearing in mind thatΓ(2) = 1,Γ(n+ 1) =n!, 1
n! ≤ Γ(1 +x)n Γ(1 +nx) ≤1.
From this it follows immediately that the functionΓ(1 +x)n/Γ(1 +nx)is strictly decreasing on(0,1].
A Geometrical Proof of a New Inequality for the Gamma
Function C. Alsina and M.S. Tomás
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References
[1] H. ALZER, On some inequalities for the gamma and psi functions, Math of Comp., 66(217) (1997), 373–389.
[2] H. ALZER, Sharp bounds for the ratio of q−Gamma functions, Math Nachr., 222 (2001), 5–14.
[3] E.W. WEISSTEIN, Hypergeometric function, [ONLINE: http://
mathworld.wolfram.com/HypergeometricFunction.html].