SOME GENERALIZED INEQUALITIES INVOLVING THE q-GAMMA FUNCTION J

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Volume 10 (2009), Issue 4, Article 120, 4 pp.

SOME GENERALIZED INEQUALITIES INVOLVING THE q-GAMMA FUNCTION

J. K. PRAJAPAT AND S. KANT DEPARTMENT OFMATHEMATICS

CENTRALUNIVERSITY OFRAJASTHAN

16, NAVDURGACOLONY, OPPOSITEHOTELCLARKSAMER, J. L. N. MARG, JAIPUR-302017, RAJASTHAN, INDIA

[email protected] DEPARTMENT OFMATHEMATICS

GOVERNMENTDUNGARCOLLEGE

BIKANER-334001, RAJASTHAN, INDIA. [email protected]

Received 06 August, 2008; accepted 14 May, 2009 Communicated by J. Sándor

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

Key words and phrases: q-Gamma Function.

2000 Mathematics Subject Classification. 33B15.

1. INTRODUCTION ANDPRELIMINARYRESULTS

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.

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2 J. K. PRAJAPAT ANDS. KANT

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

(1.5) ψ_q(x) =−log(1−q) + logq

∞

X

i=0

q^x+i 1−q^x+i. In particular

q→1lim⁻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:

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 Theorem 1.2 and Theorem 1.3 respectively.

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. Let a ≥ b > 0, c, d be positive real numbers such that bc ≥ ad > 0 and ψ(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 of Γ_q(x), given by Theorem 1.4, below:

Theorem 1.4. Letx ∈ [0,1]andq ∈ (0,1).Ifa ≥ b > 0, c, dare positive real numbers with bc≥ad >0andψ_q(b+ax)>0,then

(1.9) [Γq(a)]^c

[Γ_b(b)]^d ≤ [Γq(a+bx)]^c

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

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

Then

(1.10) ψq(aα+bx)≥ψq(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).

J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 120, 4 pp. http://jipam.vu.edu.au/

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INEQUALITIESINVOLVING THEq-GAMMAFUNCTION 3

Since for 0 < q < 1, we have logq < 0. In addition, for a ≥ 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, fora≥bandx∈[α,∞),we have(1−q^aα+bx+i)>0,(1−q^bα+ax+i)>0and q^(a−b)α ≥q^(a−b)x.Hence

ψ_q(aα+bx)≤ψ_q(bα+ax) x∈[α,∞).

which completes the proof.

Lemma 1.6. Letx ∈ [0, α], α > 0andq ∈ (0,1). Ifa, b, c, dare positive real numbers such thata ≥band[bc≥ad, ψq(bα+ax)>0]or[bc≤ad, ψq(aα+bx)<0],we have

(1.12) bcψ_q(aα+bx)−adψ_q(bα+ax)≥0.

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

≤bcψ_q(aα+bx).

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

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

This proves Lemma 1.6.

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

Lemma 1.7. Letq ∈ (0,1)andx ∈ [α,∞), α > 0.Ifa, b, c, dare 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.

2. MAINRESULTS

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

Theorem 2.1. For allq∈(0,1), x∈[0, α], α >0and positive real numbersa, b, c, dsuch that a≥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

[Γ_q(bα+ax)]^d,

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

g⁰(x) =bcΓ⁰_q(aα+bx)

Γq(aα+bx) −adΓ⁰_q(bα+ax) Γq(bα+ax)

=bcψ_q(aα+bx)−adψ_q(bα+ax).

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4 J. K. PRAJAPAT ANDS. KANT

Thus using Lemma 1.6, 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], and this is equivalent to

[Γ_q(aα)]^c

[Γq(bα)]^d ≤ [Γ_q(aα+bx)]^c

[Γq(bα+ax)]^d ≤[Γ_q{(a+b)α}]^c−d.

This completes the proof of Theorem 2.1.

Theorem 2.2. For allq ∈ (0,1), x ∈ [α,∞), α > 0and positive real numbersa, 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) [Γq(aα+bx)]^c

[Γ_q(bα+ax)]^d ≤ [Γq(aα+by)]^c

[Γ_q(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) Takingα= 1, Theorem 2.1 and Theorem 2.2 yield the results obtained by Mansour [3].

(ii) Takingα = 1andq→ 1⁻, Theorem 2.1 and Theorem 2.2 yield the results obtained by Shabani [5].

REFERENCES

[1] C. ALSINAANDM.S. THOMAS, A geometrical proof of a new inequality for the gamma function, J. Inequal. 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 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. Inequal. 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].