Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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SOME INEQUALITIES FOR THE GAMMA FUNCTION
ARMEND SH. SHABANI
Department of Mathematics University of Prishtina
Avenue "Mother Theresa", 5 Prishtine 10000, Kosova-UNMIK
EMail:armend_shabani@hotmail.com
Received: 04 May, 2007
Accepted: 15 May, 2007
Communicated by: A. Laforgia 2000 AMS Sub. Class.: 33B15.
Key words: Euler gamma function, Inequalities.
Abstract: In this paper are established some inequalities involving the Euler gamma func- tion. We use the ideas and methods that were used by J. Sándor in his paper [2].
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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Contents
1 Introduction 3
2 Main Results 4
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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1. Introduction
The Euler gamma functionΓ(x)is defined forx >0by Γ(x) =
Z ∞
0
e−ttx−1dt.
The Psi or digamma function, the logarithmic derivative of the gamma function is defined by
ψ(x) = Γ0(x)
Γ(x), x >0.
C. Alsina and M.S. Tomás in [1] proved the following double inequality:
Theorem 1.1. For allx∈[0,1]and all nonnegative integersn, the following double inequality is true:
(1.1) 1
n! ≤ Γ(1 +x)n Γ(1 +nx) ≤1.
Using the series representation of ψ(x), J. Sándor in [2] proved the following generalized result of (1.1):
Theorem 1.2. For alla≥1and allx∈[0,1], one has:
(1.2) 1
Γ(1 +a) ≤ Γ(1 +x)a Γ(1 +ax) ≤1.
In this paper, using the series representation ofψ(x)and ideas used in [2] we will establish some double inequalities involving the gamma function, "similar" to (1.2).
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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2. Main Results
In order to establish the proof of the theorems, we need the following lemmas:
Lemma 2.1. If x > 0,then the digamma function ψ(x) = Γ
0(x)
Γ(x) has the following series representation
(2.1) ψ(x) =−γ+ (x−1)
∞
X
k=0
1
(k+ 1)(k+x), whereγ is the Euler’s constant.
Proof. See [3].
Lemma 2.2. Letx ∈ [0,1]anda, bbe two positive real numbers such that a ≥ b.
Then
(2.2) ψ(a+bx)≥ψ(b+ax).
Proof. It is easy to verify thata+bx > 0, b+ax >0.Then by (2.1) we obtain:
ψ(a+bx)−ψ(b+ax) = (a+bx−1)
∞
X
k=0
1
(k+ 1)(a+bx+k)
−(b+ax−1)
∞
X
k=0
1
(k+ 1)(b+ax+k)
=
∞
X
k=0
1 k+ 1
a+bx−1
a+bx+k − b+ax−1 b+ax+k
=
∞
X
k=0
(a−b)(1−x)
(a+bx+k)(b+ax+k) ≥0.
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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Alternative proof of Lemma2.2. Letx >0, y >0andx≥y.Then ψ(x)−ψ(y) = (x−1)
∞
X
k=0
1
(k+ 1)(x+k)−(y−1)
∞
X
k=0
1 (k+ 1)(y+k)
=
∞
X
k=0
1 k+ 1
x−1
x+k − y−1 y+k
=
∞
X
k=0
(x−y)
(x+k)(y+k) ≥0.
Soψ(x)≥ψ(y).
In our case: sincea+bx >0, b+ax >0it is easy to verify that forx∈[0,1], a≥ b >0we havea+bx ≥b+ax,soψ(a+bx)≥ψ(b+ax).
Lemma 2.3. Let x ∈ [0,1], a, b (a ≥ b) be two positive real numbers such that ψ(b+ax)>0. Letc, dbe two given positive real numbers such thatbc ≥ ad >0.
Then
(2.3) bcψ(a+bx)−adψ(b+ax)≥0.
Proof. Sinceψ(b +ax) > 0, by (2.2) it is clear that ψ(a +bx) > 0. Now, since bc≥ad,using Lemma2.2, we have:
bcψ(a+bx)≥adψ(a+bx)≥adψ(b+ax).
Sobcψ(a+bx)−adψ(b+ax)≥0.
Theorem 2.4. Letf be a function defined by
f(x) = Γ(a+bx)c Γ(b+ax)d,
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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wherex ∈[0,1], a≥ b > 0, c, d are positive real numbers such that: bc ≥ad > 0 and ψ(b +ax) > 0. Then f is an increasing function on [0,1],and the following double inequality holds:
Γ(a)c
Γ(b)d ≤ Γ(a+bx)c
Γ(b+ax)d ≤ Γ(a+b)c Γ(a+b)d. Proof. Letg(x)be a function defined byg(x) = logf(x).Then:
g(x) =clog Γ(a+bx)−dlog Γ(b+ax).
So
g0(x) =bcΓ0(a+bx)
Γ(a+bx) −adΓ0(b+ax)
Γ(b+ax) =bcψ(a+bx)−adψ(b+ax).
Using (2.3), we have g0(x) ≥ 0. It means that g(x) is increasing on [0,1]. This implies thatf(x)is increasing on[0,1].
So forx∈[0,1]we havef(0) ≤f(x)≤f(1)or Γ(a)c
Γ(b)d ≤ Γ(a+bx)c
Γ(b+ax)d ≤ Γ(a+b)c Γ(a+b)d. This concludes the proof of Theorem2.4.
In a similar way, it is easy to prove the following lemmas and theorems.
Lemma 2.5. Letx≥1anda, bbe two positive real numbers such thatb≥a. Then ψ(a+bx)≥ψ(b+ax).
Inequalities for the Gamma Function Armend Sh. Shabani vol. 8, iss. 2, art. 49, 2007
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Lemma 2.6. Letx ≥ 1, a, b(b ≥ a)be two positive real numbers such thatψ(b+ ax)>0andc, dbe any two given real numbers such thatbc≥ad >0.Then
bcψ(a+bx)−adψ(b+ax)≥0.
Theorem 2.7. Letf be a function defined by
f(x) = Γ(a+bx)c Γ(b+ax)d,
wherex ≥ 1, b ≥a > 0, c, dare positive real numbers such thatbc ≥ ad > 0and ψ(b+ax)>0.Thenf is an increasing function on[1,+∞).
Lemma 2.8. Let x ∈ [0,1], a, b (a ≥ b) be two positive real numbers such that ψ(a+bx)<0andc, dbe any two given real numbers such thatad≥bc >0.Then
bcψ(a+bx)−adψ(b+ax)≥0.
Using Lemmas2.2and2.8, and the methods we used in Theorem2.4, the follow- ing theorem can be proved:
Theorem 2.9. Letf be a function defined by
f(x) = Γ(a+bx)c Γ(b+ax)d,
wherex ∈ [0,1], a ≥ b > 0, c, dare positive real numbers such thatad ≥ bc > 0 andψ(a+bx)<0.Thenf is an increasing function on[0,1].
Lemma 2.10. Letx≥1, a, b(b ≥a)be two positive real numbers such thatψ(a+ bx)<0andc, dbe any two given real numbers such thatad≥bc >0.Then
bcψ(a+bx)−adψ(b+ax)≥0.
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Using Lemmas2.5 and2.10, and the methods we used in Theorem2.4, the fol- lowing theorem can be proved:
Theorem 2.11. Letf be a function defined by
f(x) = Γ(a+bx)c Γ(b+ax)d,
wherex >1, b ≥ a > 0, c, dare positive real numbers such thatad ≥ bc >0and ψ(a+bx)<0.Thenf is an increasing function on[1,+∞).
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References
[1] C. ALSINA AND M.S. TOMÁS, A geometrical proof of a new inequality for the gamma function, J. Ineq. Pure Appl. Math., 6(2) (2005), Art. 48. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=517].
[2] J. SÁNDOR, A note on certain inequalities for the gamma function, J. Ineq. Pure Appl. Math., 6(3) (2005), Art. 61. [ONLINE:http://jipam.vu.edu.au/
article.php?sid=534].
[3] E.T. WHITTAKERANDG.N. WATSON, A Course of Modern Analysis, Camb.
Univ. Press, 1996.
[4] W. RUDIN, Principles of Mathematical Analysis, New York, McGraw-Hill, 1976.