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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:[email protected] EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 085-05

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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

http://jipam.vu.edu.au

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^−tt^x−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 geometrical argument. From now on for any r > 0,p, n ≥ 1we will consider the notation:

D_k·k^n,r

p ={(x₁, . . . , x_n)∈Rⁿ/k(x₁, . . . , x_n)k_p < r}

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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

D_k·k^n,r

p

= 2ⁿ Γ

1 + ¹_pn

Γ

1 + ⁿ_p rⁿ.

Proof. Forn = 1,D^1,r_k·k

p is the interval(−r, r), whose measure is2r, i.e., 2r= 2

Γ

1 + ¹_p Γ

1 + ¹_pr

and (1) holds. By induction, let us assume that (1) holds forn−1. Then we note that|x₁|^p+· · ·+|x_n|^p < r^p is equivalent to|x₁|^p+· · ·+|xn−1|^p < r^p− |x_n|^p and by virtue of the induction hypothesis we have

Volume

D^n,r_k·k

p

= Z

D^n,r_k·k

p

dx1. . . dxn

= 2 Z r

0

Z

Dn−1,(rp−|xn|p)1/p k·kp

dx₁. . . dxn−1

! dx_n

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= 2 Z r

0

2ⁿ⁻¹ Γ

1 + ¹_p

n−1

Γ

1 + ⁿ⁻¹_p (r^p−x^p_n)ⁿ⁻¹^p dx_n

= 2ⁿ Γ

1 + ¹_pn−1

Γ

1 + ⁿ⁻¹_p rⁿ Z 1

0

(1−z^p)ⁿ⁻¹^p dz,

wherez =x_n/r.

If we considerF(a, b, c, z)the first hypergeometric function (see [3]), then Z

(1−z^p)ⁿ⁻¹^p dz =zF 1

p,−n−1

p ,1 + 1 p, zⁿ

and by well-known properties of the hypergeometric function we deduce:

Volume D_k·k^n,r

p

= 2ⁿ Γ

1 + ¹_p

n−1

Γ

1 + ⁿ⁻¹_p rⁿF 1

p,−n−1 p ,1 + 1

p,1

= 2ⁿ Γ

1 + ¹_pn−1

Γ

1 + ⁿ⁻¹_p rⁿ Γ

1 + ¹_p Γ

1 + ⁿ⁻¹_p Γ

1 + ⁿ_p

= 2ⁿ Γ

1 + ¹_pn

Γ

1 + ⁿ_prⁿ.

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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 D^n,1_k·k

1 ⊆D_k·k^n,1

p ⊆D_k·k^n,1

∞, we deduce

Volume

D_k·k^n,1

1

≤Volume

D_k·k^n,1

p

≤Volume

D^n,1_k·k

∞

,

so by Lemma1:

2ⁿ Γ(2)ⁿ

Γ(n+ 1) ≤2ⁿ Γ

1 + ¹_p

n

Γ

1 + ⁿ_p ≤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].

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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].