NEW INEQUALITIES FOR SOME SPECIAL AND q-SPECIAL FUNCTIONS

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Inequalities for Special andq-Special Functions Mouna Sellami, Kamel Brahim

and Néji Bettaibi vol. 8, iss. 2, art. 47, 2007

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NEW INEQUALITIES FOR SOME SPECIAL AND q-SPECIAL FUNCTIONS

MOUNA SELLAMI KAMEL BRAHIM

Institut Préparatoire aux Études d’Ingénieur de El Manar Institut Préparatoire aux Études d’Ingénieur de Tunis

Tunis, Tunisia Tunis, Tunisia

EMail:mouna.sellami@ipeimt.rnu.tn EMail:kamel.brahim@ipeit.rnu.tn

NÉJI BETTAIBI

Institut Préparatoire aux Études d’Ingénieur de Mounastir, 5000 Mounastir, Tunisia.

EMail:Neji.Bettaibi@ipein.rnu.tn

Received: 14 February, 2007

Accepted: 25 May, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 33B15, 33D05.

Key words: Gamma function, Beta function, q-Gamma function, q-Beta function, q-Zeta function.

Abstract: In this paper, we give new inequalities involving some special (resp. q-special) functions, using their integral (resp. q-integral) representations and a technique developed by A. McD. Mercer in [11]. These inequalities generalize those given in [1], [2], [7] and [11].

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1. Introduction and Preliminaries

In [1], Alsina and M. S. Tomas studied a very interesting inequality involving the Gamma function and they proved the following double inequality

(1.1) 1

n! ≤ Γ(1 +x)ⁿ

Γ(1 +nx) ≤1, x∈[0,1], n∈N, by using geometric method.

In view of the interest in this type of inequalities, many authors extended this result to more general cases either for the classical Gamma function or the basic one, by using geometric or analytic approaches (see [2], [7], [12]).

In [11], A. McD. Mercer, developed a very interesting technique which was the source of some inequalities involving the Gamma, Beta and Zeta functions.

He considered a positive linear functionalLdefined on a subspaceC^∗(I)ofC(I) (the space of continuous functions onI), whereIis the interval(0, a)witha >0or equal to+∞, and he proved the following result:

Theorem 1.1. Forf, ginC^∗(I)such thatf(x)→0,g(x)→0asx→0⁺and ^f_g is strictly increasing, put

φ=gL(f) L(g)

and letF be defined on the ranges off andg such that the compositionsF(f)and F(g)each belong toC^∗(I).

a) IfF is convex then

(1.2) L[F(f)]≥L[F(φ)].

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b) IfF is concave then

(1.3) L[F(f)]≤L[F(φ)].

In this paper, using the previous theorem, we obtain some generalizations of inequalities involving some special andq-special functions.

Note that forα ∈R, the function

F(t) =t^α

is convex ifα <0orα >1and concave if0< α <1.

So, forf andgsatisfying the conditions of the previous theorem, we have:

L(f^α)> L(φ^α) if α <0 or α >1 and L(f^α)< L(φ^α) if 0< α <1.

Substituting forφthis reads:

[L(g)]^α

L(g^α) > (resp. <)[L(f)]^α L(f^α) ,

if α < 0 or α > 1 (resp. 0 < α < 1). In particular, if we take f(x) = x^β and g(x) =x^δwithβ > δ >0, we obtain the following useful inequality:

(1.4) [L(x^δ)]^α

L(x^αδ) ≷ [L(x^β)]^α L(x^αβ) ,

where, we follow the notations of [11], and ≷ correspond to the case (α < 0 or α >1) and (0< α <1) respectively.

Throughout this paper, we will fixq ∈]0,1[and we will follow the terminology and notation of the book by G. Gasper and M. Rahman [4]. We denote, in particular,

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fora∈C

[a]_q = 1−q^a

1−q , (a;q)_n=

n−1

Y

k=0

(1−aq^k), n= 1,2, . . . ,∞.

Theq-Jackson integrals from0toaand from0to∞are defined by (see [5]) (1.5)

Z a

0

f(x)d_qx= (1−q)a

∞

X

n=0

f(aqⁿ)qⁿ,

(1.6)

Z ∞

0

f(x)d_qx= (1−q)

∞

X

n=−∞

f(qⁿ)qⁿ, provided the sums converge absolutely.

Theq-Jackson integral in a generic interval[a, b]is given by (see [5]) (1.7)

Z b

a

f(x)d_qx= Z b

0

f(x)d_qx− Z a

0

f(x)d_qx.

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2. The Gamma Function

Theorem 2.1. Letf be the function defined by

(2.1) f(x) =

Γ⁽²ⁿ⁾(1 +x)α

Γ⁽²ⁿ⁾(1 +αx)

then for all0< α <1(resp.α >1)f is increasing (resp. decreasing) on(0,∞).

Proof. First, we recall that the Gamma function is infinitely differentiable on]0,+∞[

and we have

∀x∈]0,+∞[, ∀n ∈N, Γ⁽ⁿ⁾(x) = Z ∞

0

t^x−1[Log(t)]ⁿe^−tdt.

Now, we consider the subspaceC^∗(I)obtained fromC(I)by requiring its members to satisfy:

(i) w(x) = O(x^θ) (for any θ >−1) as x→0, (ii) w(x) = O(x^ϕ) (for any finite ϕ) as x→+∞.

Forw∈C^∗(I), we define

(2.2) L(w) =

Z ∞

0

w(x)(Log(x))²ⁿe^−xdx.

The linear functionalLis well-defined onC^∗(I)and it is positive.

Then, by applying the inequality (1.4), we obtain forβ > δ >0, (2.3)

Γ⁽²ⁿ⁾(1 +δ)α

Γ⁽²ⁿ⁾(1 +αδ) ≷

Γ⁽²ⁿ⁾(1 +β)α

Γ⁽²ⁿ⁾(1 +αβ) . This completes the proof.

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In particular, we have the following result, which generalizes inequality (4.1) of [11].

Corollary 2.2. For allx∈[0,1]we have:

(2.4)

Γ⁽²ⁿ⁾(2)α

Γ⁽²ⁿ⁾(1 +α) ≤

Γ⁽²ⁿ⁾(1 +x)α

Γ⁽²ⁿ⁾(1 +αx) ≤

Γ⁽²ⁿ⁾(1)^α−1

if α≥1 and

(2.5)

Γ⁽²ⁿ⁾(1)^α−1

≤

Γ⁽²ⁿ⁾(1 +x)α

Γ⁽²ⁿ⁾(1 +αx) ≤

Γ⁽²ⁿ⁾(2)α

Γ⁽²ⁿ⁾(1 +α) if 0< α≤1.

Takingn = 0, one obtains:

Corollary 2.3. For allx∈[0,1],

(2.6) 1

Γ(1 +α) ≤ [Γ(1 +x)]^α

Γ(1 +αx) ≤1, if α≥1, and

(2.7) 1≤ [Γ(1 +x)]^α

Γ(1 +αx) ≤ 1

Γ(1 +α), if 0< α≤1.

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3. The q-Gamma Function

Jackson [5] defined aq-analogue of the Gamma function by (3.1) Γ_q(x) = (q;q)∞

(q^x;q)∞

(1−q)^1−x, x6= 0,−1,−2, . . . . It is well known that it satisfies

(3.2) Γ_q(x+ 1) = [x]_qΓ_q(x), Γ_q(1) = 1 and lim

q→1⁻Γ_q(x) = Γ(x), <(x)>0.

It has the followingq-integral representation (see [8])

(3.3) Γq(s) =

Z _1−q¹

0

t^s−1E_q^−qtdqt, where

(3.4) E_q^z =₀ ϕ₀(−;−;q,−(1−q)z) =

∞

X

n=0

qⁿ⁽ⁿ⁻¹⁾² (1−q)ⁿ (q;q)n

zⁿ = (−(1−q)z;q)∞, is aq-analogue of the exponential function (see [4] and [6]).

In [3], the authors proved that Γ_q is infinitely differentiable on ]0,+∞[and we have

(3.5) ∀x∈]0,+∞[, ∀n∈N, Γ⁽ⁿ⁾_q (x) = Z _1−q¹

0

t^x−1[Log(t)]ⁿE_q^−qtdt.

Now, we are able to state aq-analogue of Theorem2.1, and give generalizations of some inequalities studied in [7].

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Theorem 3.1. Letf be the function defined by

(3.6) f(x) =

h

Γ⁽²ⁿ⁾q (1 +x)iα

Γ⁽²ⁿ⁾q (1 +αx)

then for all0< α <1(resp.α >1)f is increasing (resp. decreasing) on(0,∞).

Proof. We considerI =

0,_1−q¹

and the subspaceC^∗(I) obtained fromC(I) by requiring its members to satisfy:

(i) w(x) = O(x^θ) (for any θ >−1) as x→0, (ii) w(x) = O(1) as x→ _1−q¹ .

(3.7) L(w) =

Z _1−q¹

0

w(x)(Log(x))²ⁿE_q^−qxdqx.

Lis well-defined onC^∗(I)and it is a positive linear functional onC^∗(I).

From the inequality (1.4) and the relation (3.5), we obtain forβ > δ >0 (3.8)

h

Γ⁽²ⁿ⁾q (1 +δ)iα

Γ⁽²ⁿ⁾q (1 +αδ) ≷ h

Γ⁽²ⁿ⁾q (1 +β)iα

Γ⁽²ⁿ⁾q (1 +αβ) , which achieves the proof.

In particular, we have the following result.

Corollary 3.2. For allx∈[0,1]we have

(3.9) h

Γ⁽²ⁿ⁾q (2)iα

Γ⁽²ⁿ⁾q (1 +α)

≤ h

Γ⁽²ⁿ⁾q (1 +αx)

≤

Γ⁽²ⁿ⁾_q (1)^α−1

, if α ≥1

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and

(3.10)

Γ⁽²ⁿ⁾_q (1)^α−1

≤ h

Γ⁽²ⁿ⁾q (1 +αx) ≤ h

Γ⁽²ⁿ⁾q (2)iα

Γ⁽²ⁿ⁾q (1 +α), if 0< α≤1.

Corollary 3.3. For allx∈[0,1],

(3.11) 1

Γ_q(1 +α) ≤ [Γ_q(1 +x)]^α

Γ_q(1 +αx) ≤1, if α ≥1, and

(3.12) 1≤ [Γq(1 +x)]^α

Γ_q(1 +αx) ≤ 1

Γ_q(1 +α), if 0< α≤1.

Proof. By takingn = 0in Corollary3.2we obtain the inequalities (3.11) and (3.12).

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4. The q-Beta function

Theq-Beta function is defined by (see [4], [8]) (4.1) Bq(t, s) =

Z 1

0

x^t−1 (xq;q)_∞ (xq^s;q)∞

dqx, <(s)>0,<(t)>0 and we have

(4.2) B_q(t, s) = Γ_q(t)Γ_q(s)

Γ_q(t+s) .

Since B_q is a q-analogue of the classical Beta function, we can see the following results as generalizations of those given in [11].

Theorem 4.1. Fors >0, letf be the function defined by

(4.3) f(x) = [B_q(1 +x, s)]^α

B_q(1 +αx, s). If0< α <1, f is increasing on[0,+∞[.

Ifα >1f is decreasing on[0,+∞[.

Proof. We consider the intervalI = (0,1)and the subspace C^∗(I) obtained from C(I)by requiring its members to satisfy:

(i) w(x) = O(x^θ) (for any θ >−1) as x→0, (ii) w(x) = O(1) as x→1.

Fors >0, we put forw∈C^∗(I),

(4.4) L(w) =

Z 1

0

w(x)(xq;q)∞

(xq^s;q)_∞d_qx.

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It is easy to see thatLis well-defined onC^∗(I)and it is a positive linear functional onC^∗(I).

Then, from the inequality (1.4), we obtain forβ > δ >0

(4.5) [B_q(1 +δ, s)]^α

B_q(1 +αδ, s) ≷ [B_q(1 +β, s)]^α B_q(1 +αβ, s) . This achieves the proof.

Corollary 4.2. For allx∈[0,1],s >0

(4.6) [α+s]_q

[α]q[s]^α_q[s+ 1]^α_qBq(α, s) ≤ [B_q(1 +x, s)]^α

Bq(1 +αx, s) ≤ 1

[s]^α−1_q , if α≥1.

Proof. It is a consequence of the previous theorem and the relations:

B_q(1, s) = 1

[s]_q, B_q(2, s) = 1 [s]_q[s+ 1]_q and

B_q(1 +α, s) = [α]_q

[α+s]_qB_q(α, s).

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5. The q- Zeta Function

Forx >0, we put

α(x) = Log(x) Log(q) −E

Log(x) Log(q)

and

{x}q = [x]_q q^x+α([x]^q⁾, whereE_Log(x)

Log(q)

is the integer part of ^Log(x)_Log(q).

In [3], the authors defined theq-Zeta function as follows

(5.1) ζ_q(s) =

∞

X

n=1

1 {n}^s_q =

∞

X

n=1

q^(n+α([n]^q^))s [n]^s_q .

They proved that it is aq-analogue of the classical Riemann Zeta function and in the additional assumption ^Log(1−q)_Log(q) ∈Z, we have for alls∈Csuch that<(s)>1,

ζ_q(s) = 1 Γe_q(s)

Z ∞

0

t^s−1Z_q(t)d_qt, where for allt >0,

Z_q(t) =

∞

X

n=1

e^−{n}_q ^q^t and Γe_q(t) = Γ_q(t)(−q^t,−q^1−t;q)_∞ (−q,−1;q)∞

.

Now, we consider the subspaceC^∗(I)obtained fromC(I)by requiring its members to satisfy:

(i) w(x) = O(x^θ) (for any θ >−1) as x→0,

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(ii) w(x) = O(x^ϕ) (for any finite ϕ) as x→+∞.

(5.2) L(w) =

Z ∞

0

w(x)Z_q(x)d_qx.

Lis a positive linear functional onC^∗(I). So, by application of the inequality (1.4), we obtain for allβ > δ >0,

h

eΓ_q(1 +δ)ζ_q(1 +δ)iα

Γe_q(1 +αδ)ζ_q(1 +αδ) ≷ h

eΓ_q(1 +β)ζ_q(1 +β)iα

eΓ_q(1 +αβ)ζ_q(1 +αβ).

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References

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