Recently, Shabani [4, Theorem 2.4] established some inequalities involving the gamma function

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Volume 9 (2008), Issue 1, Article 18, 4 pp.

SOME INEQUALITIES FOR THE q-GAMMA FUNCTION

TOUFIK MANSOUR DEPARTMENT OFMATHEMATICS

UNIVERSITY OFHAIFA

31905 HAIFA, ISRAEL. [email protected]

Received 29 July, 2007; accepted 29 January, 2008 Communicated by J. Sándor

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 the q-gamma function.

Key words and phrases: q-gamma function, Inequalities.

2000 Mathematics Subject Classification. 33B15.

1. INTRODUCTION

The Euler gamma functionΓ(x)is defined forx >0by Γ(x) =

Z ∞

0

e^−te^x−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 andx ∈ [0,1], see [3]. Recently, Shabani [4] using the series representation of the functionψ(x)and the ideas used in [3] established some double inequalities involving the

The author would like to thank Armend Shabani for reading previous version of the present paper.

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2 TOUFIKMANSOUR

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,

for all x ∈ [0,1], a ≥ b > 0, c, d are positive real numbers such that bc ≥ ad > 0, and ψ(b+ax)>0.

In this paper we give the q-inequalities of the above results by using similar techniques 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 function as (0< q <1)

ψq(x) = d

dxlog Γq(x), where theq-gamma functionΓq(x)is defined by (0< q <1)

Γ_q(x) = (1−q)^1−x

∞

Y

i=1

1−qⁱ 1−q^x+i.

Many properties of theq-gamma function were derived by Askey [2]. The explicit form ofq-psi functionψ_q(x)is

(1.2) ψ_q(x) =−log(1−q) + logq

∞

X

i=0

q^x+i 1−q^x+i.

In this paper we extend (1.1) to the case of Γq(x). In particular, by using the facts that limq→1⁻Γq(x) = Γ(x)andlimq→1⁻ψq(x) =ψ(x)we obtain all the results of Shabani [4].

2. MAINRESULTS

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 that a≥b. Then

ψ_q(a+bx)≥ψ_q(b+ax).

Proof. Clearly,a+bx, b+ax >0. The series presentation ofψ_q(x), see (1.2), gives ψ_q(a+bx)−ψ_q(b+ax) = logq

∞

X

i=0

q^a+bx+i

1−q^a+bx+i − q^b+ax+i 1−q^b+ax+i

= logq

∞

X

i=0

qⁱ(q^a+bx−q^b+ax) (1−q^a+bx+i)(1−q^b+ax+i)

= logq

∞

X

i=0

q^b+bx+i(q^a−b−q^(a−b)x) (1−q^a+bx+i)(1−q^b+ax+i).

Since0 < q < 1we have that logq < 0. In addition, sincea ≥ b we get thatq^a−b ≤ q^(a−b)x. Hence,

ψ_q(a+bx)−ψ_q(b+ax)≥0,

which completes the proof.

Lemma 2.2. Letx ∈ [0,1], q ∈ (0,1), a, bbe any two positive real numbers such that a ≥ b andψ_q(b+ax)>0. Letc, dbe any two positive real numbers such thatbc≥ad >0. Then

bcψ_q(a+bx)−adψ_q(b+ax)≥0.

J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 18, 4 pp. http://jipam.vu.edu.au/

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INEQUALITIES FOR THEq-GAMMAFUNCTION 3

Proof. Lemma 2.1 together withψ_q(b+ax) > 0give that ψ_q(a+bx) > 0. Thus Lemma 2.1 obtains

bcψ_q(a+bx)≥adψ_q(a+bx)≥adψ_q(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 withbc≥ad >0 andψ_q(b+ax)>0. Then

Γq(a)^c

Γ_q(b)^d ≤ Γq(a+bx)^c

Γ_q(b+ax)^d ≤ Γq(a+b)^c Γ_q(a+b)^d. Proof. Letf(x) = ^Γ_Γ^q^(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)

=bcΓ⁰_q(a+bx)

Γ_q(a+bx) −adΓ⁰_q(b+ax) Γ(b+ax)

=bcψ_q(a+bx)−adψ_q(b+ax).

Thus, Lemma 2.2 givesg⁰(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

Γ_q(a)^c

Γ_q(b)^d ≤ Γ_q(a+bx)^c

Γ_q(b+ax)^d ≤ Γ_q(a+b)^c Γ_q(a+b)^d,

as requested.

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 withb ≥a. Then ψ_q(a+bx)≥ψ_q(b+ax).

Lemma 2.5. Let x ≥ 1, q ∈ (0,1), a, b be any two positive real numbers with b ≥ a and ψ_q(b+ax)>0, andc, dbe any two real numbers such thatbc≥ad >0. Then

Using similar techniques to the ones in the proof of Theorem 2.3 with Lemmas 2.4 and 2.5, instead of Lemmas 2.1 and 2.2, we can prove the following result.

Theorem 2.6. Letx≥ 1,q ∈(0,1),a, bbe any two positive real numbers withb≥ a >0and ψ_q(b+ax)>0, andc, dbe any two real numbers such thatbc ≥ad > 0. Then ^Γ_Γ^q^(a+bx)^c

q(b+ax)^d is an increasing function on[1,+∞).

In addition, similar arguments as in the proof of Lemma 2.2 will obtain the following lemmas.

Lemma 2.7. Letx ∈ [0,1], q ∈ (0,1), a, bbe any two positive real numbers witha ≥ b > 0 andψ_q(a+bx)<0, andc, dbe any two real numbers such thatad≥bc >0. Then

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4 TOUFIKMANSOUR

Lemma 2.8. Let x ≥ 1, q ∈ (0,1), a, b be any two positive real numbers with b ≥ a and ψ_q(a+bx)<0, andc, dbe any two real numbers such thatad ≥bc >0. Then

Using similar techniques to the ones in the proof of Theorem 2.3, with Lemmas 2.2 and 2.7, we obtain the following.

Theorem 2.9. Letx∈ [0,1], q ∈ (0,1),a, bbe any two positive real numbers witha ≥ b > 0 andψ_q(a+bx)<0, andc, dbe any two real numbers such thatad≥bc >0. Then ^Γ_Γ^q^(a+bx)^c

q(b+ax)^d is an increasing function on[0,1].

Using similar techniques to the ones in the proof of Theorem 2.3, with Lemmas 2.4 and 2.8, we obtain the following.

Theorem 2.10. Letx≥1,q∈(0,1),a, bbe any two positive real numbers withb ≥a >0and ψ_q(a+bx)<0, andc, dbe any two real numbers such thatad ≥bc > 0. Then ^Γ_Γ^q^(a+bx)^c

q(b+ax)^d is an increasing function on[1,+∞).

REFERENCES

[1] A. ALSINA AND M.S. TOMÁS, A geometrical proof of a new inequality for the gamma func- tion, J. Ineq. Pure Appl. Math., 6(2) (2005), Art. 48. [ONLINE:http://jipam.vu.edu.au/

article.php?sid=517].

[2] R. ASKEY, Theq-gamma andq-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].