Inequalities for the q-Gamma Function Toufik Mansour vol. 9, iss. 1, art. 18, 2008
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SOME INEQUALITIES FOR THE q -GAMMA FUNCTION
TOUFIK MANSOUR
Department of Mathematics University of Haifa 31905 Haifa, Israel.
EMail:toufik@math.haifa.ac.il
Received: 29 July, 2007
Accepted: 29 January, 2008
Communicated by: J. Sándor 2000 AMS Sub. Class.: 33B15.
Key words: q-gamma function, Inequalities.
Abstract: Recently, Shabani [4, Theorem 2.4] established some inequalities involving the gamma function. In this paper we present theq-analogues of these inequalities involving theq-gamma function.
Acknowledgements: The author would like to thank Armend Shabani for reading previous version of the present paper.
Inequalities for the q-Gamma Function Toufik Mansour vol. 9, iss. 1, art. 18, 2008
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Contents
1 Introduction 3
2 Main Results 5
Inequalities for the q-Gamma Function Toufik Mansour vol. 9, iss. 1, art. 18, 2008
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1. Introduction
The Euler gamma functionΓ(x)is defined forx >0by Γ(x) =
Z ∞
0
e−tex−1dt.
The psi or digamma function, the logarithmic derivative of the gamma function is defined by
ψ(x) = Γ0(x)
Γ(x), x >0.
Alsina and Tomás [1] proved that 1
n! ≤ Γ(1 +x)n Γ(1 +nx) ≤1,
for allx∈[0,1]and nonnegative integersn. This inequality can be generalized to 1
Γ(1 +a) ≤ Γ(1 +x)a Γ(1 +ax) ≤1,
for all a ≥ 1 and x ∈ [0,1], see [3]. Recently, Shabani [4] using the series rep- resentation of the functionψ(x)and the ideas used in [3] established some double inequalities involving the gamma function. In particular, Shabani [4, Theorem 2.4]
proved
(1.1) Γ(a)c
Γ(b)d ≤ Γ(a+bx)c
Γ(b+ax)d ≤ Γ(a+b)c Γ(a+b)d,
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In this paper we give theq-inequalities of the above results by using similar tech- niques to those in [4]. The main ideas of Shabani’s paper, as well as of the present one, are contained in paper [3] by Sándor. More precisely, we define theq-psi func- tion as (0< q <1)
ψq(x) = d
dxlog Γq(x),
where theq-gamma functionΓq(x)is defined by (0< q <1) Γq(x) = (1−q)1−x
∞
Y
i=1
1−qi 1−qx+i.
Many properties of theq-gamma function were derived by Askey [2]. The explicit form ofq-psi functionψq(x)is
(1.2) ψq(x) =−log(1−q) + logq
∞
X
i=0
qx+i 1−qx+i.
In this paper we extend (1.1) to the case of Γq(x). In particular, by using the facts thatlimq→1−Γq(x) = Γ(x)andlimq→1−ψq(x) = ψ(x) we obtain all the results of Shabani [4].
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2. Main Results
In order to establish the proof of the theorems, we need the following lemmas.
Lemma 2.1. Let x ∈ [0,1], q ∈ (0,1), and a, b be any two positive real numbers such thata≥b. Then
ψq(a+bx)≥ψq(b+ax).
Proof. Clearly,a+bx, b+ax >0. The series presentation ofψq(x), see (1.2), gives ψq(a+bx)−ψq(b+ax) = logq
∞
X
i=0
qa+bx+i
1−qa+bx+i − qb+ax+i 1−qb+ax+i
= logq
∞
X
i=0
qi(qa+bx−qb+ax) (1−qa+bx+i)(1−qb+ax+i)
= logq
∞
X
i=0
qb+bx+i(qa−b−q(a−b)x) (1−qa+bx+i)(1−qb+ax+i).
Since 0 < q < 1 we have that logq < 0. In addition, since a ≥ b we get that qa−b ≤q(a−b)x. Hence,
ψq(a+bx)−ψq(b+ax)≥0, which completes the proof.
Lemma 2.2. Letx ∈ [0,1], q ∈ (0,1), a, bbe any two positive real numbers such thata ≥ bandψq(b+ax) >0. Letc, dbe any two positive real numbers such that bc≥ad >0. Then
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Proof. Lemma2.1 together with ψq(b+ax) > 0give thatψq(a+bx) > 0. Thus Lemma2.1obtains
bcψq(a+bx)≥adψq(a+bx)≥adψq(b+ax), as required.
Now we present theq-inequality of (1.1) .
Theorem 2.3. Letx∈ [0,1], q ∈ (0,1), a ≥b > 0, c, dpositive real numbers with bc≥ad >0andψq(b+ax)>0. Then
Γq(a)c
Γq(b)d ≤ Γq(a+bx)c
Γq(b+ax)d ≤ Γq(a+b)c Γq(a+b)d. Proof. Letf(x) = ΓΓq(a+bx)c
q(b+ax)d andg(x) = logf(x). Then
g(x) =clog Γq(a+bx)−dlog Γq(b+ax), which implies that
g0(x) = d dxg(x)
=bcΓ0q(a+bx)
Γq(a+bx) −adΓ0q(b+ax) Γ(b+ax)
=bcψq(a+bx)−adψq(b+ax).
Thus, Lemma2.2 gives g0(x) ≥ 0, that is, g(x)is an increasing function on [0,1].
Therefore,f(x)is an increasing function on[0,1]. Hence, for allx∈[0,1]we have thatf(0)≤f(x)≤f(1), which is equivalent to
Γq(a)c
Γq(b)d ≤ Γq(a+bx)c
Γq(b+ax)d ≤ Γq(a+b)c Γq(a+b)d, as requested.
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Similarly as in the argument proofs of Lemmas 2.1 – 2.2 and Theorem 2.3 we obtain the following results.
Lemma 2.4. Letx ≥ 1,q ∈ (0,1), anda, bbe any two positive real numbers with b≥a. Then
ψq(a+bx)≥ψq(b+ax).
Lemma 2.5. Letx≥1,q ∈(0,1),a, bbe any two positive real numbers withb ≥a andψq(b+ax)>0, andc, dbe any two real numbers such thatbc≥ad >0. Then
bcψq(a+bx)−adψq(b+ax)≥0.
Using similar techniques to the ones in the proof of Theorem 2.3 with Lem- mas2.4and2.5, instead of Lemmas2.1and2.2, we can prove the following result.
Theorem 2.6. Let x ≥ 1, q ∈ (0,1), a, b be any two positive real numbers with b ≥ a > 0 and ψq(b + ax) > 0, and c, d be any two real numbers such that bc≥ad >0. Then ΓΓq(a+bx)c
q(b+ax)d is an increasing function on[1,+∞).
In addition, similar arguments as in the proof of Lemma 2.2will obtain the fol- lowing lemmas.
Lemma 2.7. Letx ∈ [0,1], q ∈ (0,1), a, bbe any two positive real numbers with a ≥ b > 0 and ψq(a + bx) < 0, and c, d be any two real numbers such that ad≥bc >0. Then
bcψq(a+bx)−adψq(b+ax)≥0.
Lemma 2.8. Letx≥1,q ∈(0,1),a, bbe any two positive real numbers withb ≥a andψ (a+bx)<0, andc, dbe any two real numbers such thatad≥bc >0. Then
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Using similar techniques to the ones in the proof of Theorem2.3, with Lemmas 2.2and2.7, we obtain the following.
Theorem 2.9. Let x ∈ [0,1], q ∈ (0,1), a, b be any two positive real numbers witha ≥ b > 0 andψq(a+bx) < 0, and c, dbe any two real numbers such that ad≥bc >0. Then ΓΓq(a+bx)c
q(b+ax)d is an increasing function on[0,1].
Using similar techniques to the ones in the proof of Theorem2.3, with Lemmas 2.4and2.8, we obtain the following.
Theorem 2.10. Let x ≥ 1, q ∈ (0,1), a, bbe any two positive real numbers with b ≥ a > 0 and ψq(a + bx) < 0, and c, d be any two real numbers such that ad≥bc >0. Then ΓΓq(a+bx)c
q(b+ax)d is an increasing function on[1,+∞).
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References
[1] A. 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] R. ASKEY, The q-gamma and q-beta functions, Applicable Anal., 8(2) (1978/79), 125–141.
[3] 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].
[4] A. Sh. SHABANI, Some inequalities for the gamma function, J. Ineq. Pure Appl. Math., 8(2) (2007) Art. 49. [ONLINE:http://jipam.vu.edu.au/
article.php?sid=852].