SOME GENERALIZED INEQUALITIES INVOLVING THE q-GAMMA FUNCTION

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Inequalities Involving the q-Gamma Function J.K. Prajapat and S. Kant vol. 10, iss. 4, art. 120, 2009

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SOME GENERALIZED INEQUALITIES INVOLVING THE q-GAMMA FUNCTION

J. K. PRAJAPAT S. KANT

Department of Mathematics Department of Mathematics Central University of Rajasthan Government Dungar College 16, Nav Durga Colony, Opposite Hotel Clarks Amer, Bikaner-334001,

J. L. N. Marg, Jaipur-302017, Rajasthan, India Rajasthan, India.

EMail:jkp_0007@rediffmail.com EMail:drskant.2007@yahoo.com

Received: 06 August, 2008

Accepted: 14 May, 2009

Communicated by: J. Sándor 2000 AMS Sub. Class.: 33B15.

Key words: q-Gamma Function.

Abstract: In this paper we establish some generalized double inequalities involving theq- gamma function.

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1. Introduction and Preliminary Results

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

(1.1) Γ(x) =

Z ∞

0

e^−tt^x−1dt,

and the Psi (or digamma) function is defined by

(1.2) ψ(x) = Γ⁰(x)

Γ(x) (x >0).

Theq-psi function is defined for0< q <1,by

(1.3) ψ_q(x) = d

dxlog Γ_q(x),

where theq-gamma functionΓ_q(x)is defined by(0< q <1)

(1.4) 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 of theq-psi functionψ_q(x)is

(1.5) ψq(x) =−log(1−q) + logq

∞

X

i=0

q^x+i 1−q^x+i. In particular

lim

q→1⁻Γ_q(x) = Γ(x) and lim

q→1⁻ψ_q(x) =ψ(x).

For the gamma function Alsina and Thomas [1] proved the following double inequality:

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Theorem 1.1. For allx∈[0,1], and all nonnegative integersn, the following double inequality holds true

(1.6) 1

n! ≤ [Γ(1 +x)]ⁿ Γ(1 +nx) ≤1.

Sándor [4] and Shabani [5] proved the following generalizations of (1.6) given by Theorem1.2and Theorem1.3respectively.

Theorem 1.2. For alla≥1and allx∈[0,1], one has

(1.7) 1

Γ(1 +a) ≤ [Γ(1 +x)]^a Γ(1 +ax) ≤1.

Theorem 1.3. Leta ≥ b > 0, c, d be positive real numbers such thatbc ≥ ad > 0 andψ(b+ax)>0,wherex∈[0,1].Then the following double inequality holds:

(1.8) [Γ(a)]^c

[Γ(b)]^d ≤ [Γ(a+bx)]^c

[Γ(b+ax)]^d ≤[Γ(a+b)]^c−d.

Recently, Mansour [3] extended above gamma function inequalities to the case ofΓ_q(x),given by Theorem1.4, below:

Theorem 1.4. Let x ∈ [0,1] and q ∈ (0,1). If a ≥ b > 0, c, dare positive real numbers withbc≥ad >0andψ_q(b+ax)>0,then

(1.9) [Γ_q(a)]^c

[Γ_b(b)]^d ≤ [Γ_q(a+bx)]^c

[Γ_q(b+ax)]^d ≤[Γ_q(a+b)]^c−d. In our investigation we shall require the following lemmas:

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Lemma 1.5. Letq ∈ (0,1), α > 0anda, bbe any two positive real numbers such thata≥b.Then

(1.10) ψ_q(aα+bx)≥ψ_q(bα+ax) x∈[0, α], and

(1.11) ψ_q(aα+bx)≤ψ_q(bα+ax) x∈[α,∞).

Proof. By using (1.5), we have

ψ_q(aα+bx)−ψ_q(bα+ax) = logq

∞

X

i=0

q^aα+bx+i

1−q^aα+bx+i − q^bα+ax+i 1−q^bα+ax+i

= logq

∞

X

i=0

qⁱ q^aα+bx−q^bα+ax (1−q^aα+bx+i)(1−q^bα+ax+i)

= logq

∞

X

i=0

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

Since for0 < q < 1,we have logq < 0.In addition, fora ≥ b, x ∈ [0, α],we get (1−q^aα+bx+i)>0,(1−q^bα+ax+i)>0andq^(a−b)α ≤q^(a−b)x.Hence

ψ_q(aα+bx)≥ψ_q(bα+ax) x∈[0, α].

Furthermore, for a ≥ b and x ∈ [α,∞), we have (1−q^aα+bx+i) > 0, (1− q^bα+ax+i)>0andq^(a−b)α ≥q^(a−b)x.Hence

ψ_q(aα+bx)≤ψ_q(bα+ax) x∈[α,∞).

which completes the proof.

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Lemma 1.6. Let x ∈ [0, α], α > 0 and q ∈ (0,1). If a, b, c, d are positive real numbers such thata≥band[bc≥ad, ψ_q(bα+ax)>0]or[bc≤ad, ψ_q(aα+bx)<

0],we have

(1.12) bcψ_q(aα+bx)−adψ_q(bα+ax)≥0.

Proof. Sincebc≥adandψ_q(bα+ax)>0,then using (1.10), we obtain adψ_q(bα+ax)≤bcψ_q(bα+ax)

≤bcψ_q(aα+bx).

Similarly, whenbc≤adandψ_q(aα+bx)<0, we have

bcψ_q(aα+bx)≥adψ_q(aα+bx)≥adψ_q(bα+ax).

This proves Lemma1.6.

Similarly, using (1.11) and a similar proof to that above, we have the following lemma:

Lemma 1.7. Let q ∈ (0,1)and x ∈ [α,∞), α > 0. If a, b, c, d are positive real numbers such thata≥band[bc≥ad, ψ_q(bα+ax)<0]or[bc≤ad, ψ_q(aα+bx)<

0],we have

(1.13) bcψ_q(aα+bx)−adψ_q(bα+ax)≤0.

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2. Main Results

In this section we will establish some generalized double inequalities involving the q- gamma function.

Theorem 2.1. For all q ∈ (0,1), x ∈ [0, α], α > 0 and positive real numbers a, b, c, dsuch thata≥band[bc≥ad, ψ_q(bα+ax)>0]or[bc≤ad, ψ_q(aα+bx)<

0],we have

(2.1) [Γ_q(aα)]^c

[Γ_q(bα)]^d ≤ [Γ_q(aα+bx)]^c

[Γ_q(bα+ax)]^d ≤[Γ_q{(a+b)α}]^c−d. Proof. Let

(2.2) f(x) = [Γ_q(aα+bx)]^c

[Γq(bα+ax)]^d,

and assume thatg(x)is a function defined byg(x) = logf(x).Then g(x) =clog Γ_q(aα+bx)−dlog Γ_q(bα+ax), so

g⁰(x) =bcΓ⁰_q(aα+bx)

Γ_q(aα+bx) −adΓ⁰_q(bα+ax) Γ_q(bα+ax)

=bcψq(aα+bx)−adψq(bα+ax).

Thus using Lemma1.6, we haveg⁰(x)≥0.This means thatg(x)is an increasing function in[0, α],which implies that the functionf(x)is also an increasing function in[0, α],so that

f(0) ≤f(x)≤f(α), x∈[0, α],

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and this is equivalent to [Γ_q(aα)]^c

[Γ_q(bα)]^d ≤ [Γ_q(aα+bx)]^c

[Γ_q(bα+ax)]^d ≤[Γ_q{(a+b)α}]^c−d. This completes the proof of Theorem2.1.

Theorem 2.2. For all q ∈ (0,1), x ∈ [α,∞), α > 0 and positive real numbers a, b, c, dsuch thata≥band[bc≥ad, ψ_q(bα+ax)<0]or[bc≤ad, ψ_q(aα+bx)>

0],we have

(2.3) [Γ_q(aα+bx)]^c

[Γ_q(bα+ax)]^d ≤[Γ_q(a+b)α]^c−d and

(2.4) [Γ_q(aα+bx)]^c

[Γq(bα+ax)]^d ≤ [Γ_q(aα+by)]^c

[Γq(bα+ay)]^d, α < y < x.

Proof. Applying Lemma 1.7 and an argument similiar to that of Theorem 2.1, we see that the functionf(x) defined by (2.2) is a decreasing function. Therefore we have

f(x)≤f(α), x∈[α,∞), which gives the desired result.

Remark 1.

(i) Taking α = 1, Theorem 2.1 and Theorem 2.2 yield the results obtained by Mansour [3].

(ii) Taking α = 1 and q → 1⁻, Theorem 2.1 and Theorem2.2 yield the results obtained by Shabani [5].

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References

[1] C. ALSINAAND M.S. THOMAS, A geometrical proof of a new inequality for the gamma function, J. Inequal. Pure & Appl. Math., 6(2) (2005), Art. 48. [ON- LINE:http://jipam.vu.edu.au/article.php?sid=517].

[2] R. ASKEY, Theq-gamma andq-beta function, Applicable Anal., 8(2) (1978/79), 125–141.

[3] T. MANSOUR, Some inequalities for q-gamma function, J. Inequal. Pure &

Appl. Math., 9(1) (2008), Art. 18. [ONLINE:http://jipam.vu.edu.au/

article.php?sid=954].

[4] J. SÁNDOR, A note on certain inequalities for the gamma function, J. In- equal. Pure Appl. Math., 6(3) (2005), Art. 61. [ONLINE: http://jipam.

vu.edu.au/article.php?sid=534].

[5] A.S. SHABANI, Some inequalities for the gamma function, J. Inequal. Pure Appl. Math., 8(2) (2007), Art. 49. [ONLINE:http://jipam.vu.edu.au/

article.php?sid=852].