AN INEQUALITY ABOUT 3

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Inequality About3φ₂ Mingjin Wang and

Hongshun Ruan vol. 9, iss. 2, art. 48, 2008

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AN INEQUALITY ABOUT

₃

φ

₂

AND ITS APPLICATIONS

MINGJIN WANG AND HONGSHUN RUAN

Department of Information Science Jiangsu Polytechnic University Changzhou City 213164 Jiangsu Province, P.R. China.

EMail:wmj@jpu.edu.cn rhs@em.jpu.edu.cn

Received: 17 August, 2007

Accepted: 18 May, 2008

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary 26D15; Secondary 33D15.

Key words: Basic hypergeometric function3φ2;q-binomial theorem;q-Chu-Vandermonde formula; Grüss inequality.

Abstract: In this paper, we use the terminating case of theq-binomial formula, theq-Chu- Vandermonde formula and the Grüss inequality to drive an inequality about3φ2. As applications of the inequality, we discuss the convergence of someq-series involving3φ2.

Acknowledgements: The author would like to express deep appreciation to the referee for the helpful suggestions. In particular, the author thanks the referee for helping to improve the presentation of the paper.

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Inequality About3φ2

Mingjin Wang and Hongshun Ruan vol. 9, iss. 2, art. 48, 2008

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1. Statement of Main Results

q-series, which are also called basic hypergeometric series, play a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials and physics, etc. Inequalities techniques provide useful tools in the study of special functions (see [1, 6, 7, 8, 9, 10]). For example, Ito used inequalities techniques to give a sufficient condition for convergence of a specialq-series called the Jackson integral [6]. In this paper, we derive the following new inequality aboutq-series involving₃φ₂.

Theorem 1.1. Leta₁, a₂, b₁, b₂ be some real numbers such thatb_i <1fori = 1,2.

Then for all positive integersn, we have:

(1.1) ³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ b1, b2

;q,−a₁a₂qⁿ

− (a₁, a₂;q)_n (−1, b1, b2;q)n

≤λµ(−1;q)_n, where

λ= max{1, Mⁿ}, M = max

|a1|,|a₁−b₁| 1−b₁

, µ= max{1, Nⁿ}, N = max

|a2|,|a₂−b₂| 1−b₂

. Applications of this inequality are also given.

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2. Notations and Known Results

We recall some definitions, notations and known results which will be used in the proof. Throughout this paper, it is supposed that0< q <1. Theq-shifted factorials are defined as

(2.1) (a;q)₀ = 1, (a;q)_n=

n−1

Y

k=0

(1−aq^k), (a;q)∞ =

∞

Y

k=0

(1−aq^k).

We also adopt the following compact notation for multipleq-shifted factorials:

(2.2) (a1, a2, . . . , am;q)n = (a1;q)n(a2;q)n· · ·(am;q)n, wherenis an integer or∞.

Theq-binomial theorem (see [2,3,4]) is given by (2.3)

∞

X

k=0

(a;q)_kz^k

(q;q)_k = (az;q)∞

(z;q)_∞ , |z|<1.

Whena=q⁻ⁿ, wherendenotes a nonnegative integer, we have (2.4)

n

X

k=0

(q⁻ⁿ;q)_kz^k

(q;q)_k = (zq⁻ⁿ;q)_n.

Heine introduced the_r+1φ_rbasic hypergeometric series, which is defined by (2.5) r+1φ_r

a₁, a₂, . . . , a_r+1 b₁, b₂, . . . , b_r ;q, x

=

∞

X

n=0

(a₁, a₂, . . . , a_r+1;q)_nxⁿ (q, b₁, b₂, . . . , b_r;q)_n .

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Theq-Chu-Vandermonde sums (see [2,3,4]) are

(2.6) 2φ₁

a, q⁻ⁿ c ;q, q

= aⁿ(c/a;q)_n (c;q)n

and, reversing the order of summation, we have

(2.7) ₂φ₁

a, q⁻ⁿ

c ;q, cqⁿ/a

= (c/a;q)_n (c;q)n

. At the end of this section, we recall the Grüss inequality (see [5]):

(2.8)

1 b−a

Z b

a

f(x)g(x)dx− 1

b−a Z b

a

f(x)dx 1

b−a Z b

a

g(x)dx

≤ (M −m)(N −n)

4 ,

provided that f, g : [a, b] → Rare integrable on [a, b] and m ≤ f(x) ≤ M, n ≤ g(x)≤N for allx∈[a, b], wherem,M,n,N are given constants.

By simple calculus, one can prove that the discrete version of the Grüss inequality can be stated as:

ifa ≤λ_i ≤Aandb ≤µ_i ≤B, i= 1,2, . . . , n, then for all sequences(p_i)0≤i≤n

of nonnegative real numbers satisfyingPn

i=1p_i = 1, we have (2.9)

n

X

i=1

λ_iµ_ip_i−

n

X

i=1

λ_ip_i

!

·

n

X

i=1

µ_ip_i

!

≤ (A−a)(B−b)

4 ,

wherea,A,b,B are some given real constants.

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3. Proof of the Theorem

In this section, we use the terminating case of theq-binomial formula, theq-Chu- Vandermonde formula and the Grüss inequality to prove (1.1). For this purpose, we need the following lemma.

Lemma 3.1. Letaandb be two real numbers such that b < 1, and let0 ≤ t ≤ 1.

Then,

(3.1)

a−bt 1−bt

≤max

|a|,|a−b|

1−b

. Proof. Let

f(t) = a−bt

1−bt, 0≤t≤1, then

f⁰(t) = b(a−1)

(1−bt)², 0≤t≤1.

So f(t) is a monotonic function with respect to 0 ≤ t ≤ 1. Since f(0) = a and f(1) = ^a−b_1−b, (3.1) holds.

Now, we are in a position to prove the inequality (1.1).

Proof. Put

(3.2) p_k = (q⁻ⁿ;q)k(−qⁿ)^k

(q;q)_k(−1;q)_n , k= 0,1, . . . , n.

It is obvious thatp_k ≥0.

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On the other hand, using (2.4), we obtain

n

X

k=0

p_k = 1

(−1;q)_n

n

X

k=0

(q⁻ⁿ;q)_k(−qⁿ)^k (q;q)_k = 1.

Let

(3.3) λ_k = (−a1)^k(b1/a1;q)k

(b₁;q)_k , and

(3.4) µk = (−a₂)^k(b₂/a₂;q)_k (b₂;q)_k , wherek = 0,1, . . . , n.

According to the definitions ofM, N, λandµ, it is easy to see that M^k ≤λ and N^k≤µ, 0≤k ≤n.

Using the lemma, one can get for all0≤k ≤n, (3.5) |λ_k|=

a₁−b₁ 1−b1

· a₁−b₁q

1−b1q · · · a₁−b₁q^k−1 1−b1q^k−1

≤M^k ≤λ and

(3.6) |µ_k|=

a₂−b₂

1−b₂ · a₂−b₂q

1−b₂q · · · a₂−b₂q^k−1 1−b₂q^k−1

≤N^k ≤µ.

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Substitution of (3.2), (3.3), (3.4), (3.5) and (3.6) into (2.9), gives (3.7)

n

X

k=0

(q⁻ⁿ;q)_k(−qⁿ)^k

(q;q)_k(−1;q)_n · (−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k ·(−a₂)^k(b₂/a₂;q)_k (b₂;q)_k

−

n

X

k=0

(q;q)_k(−1;q)_n · (−a₁)^k(b₁/a₁;q)_k (b₁;q)_k

×

n

X

k=0

(q;q)_k(−1;q)_n ·(−a₂)^k(b₂/a₂;q)_k (b₂;q)_k

≤λµ.

Using (2.5) and (2.7), we get

n

X

k=0

(q;q)_k(−1;q)_n · (−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k · (−a₂)^k(b₂/a₂;q)_k (b₂;q)_k (3.8)

= 1

(−1;q)_n

n

X

k=0

(q⁻ⁿ, b1/a1, b2/a2;q)k

(q, b₁, b₂;q)_k (−a₁a₂qⁿ)^k

= 1

(−1;q)_n³φ2

b₁/a₁, b₂/a₂, q⁻ⁿ

b₁, b₂ ;q,−a1a2qⁿ

,

(3.9)

n

X

k=0

(q⁻ⁿ;q)_k(−qⁿ)^k (q;q)k(−1;q)n

·(−a₁)^k(b₁/a₁;q)_k (b1;q)k

= (a₁;q)_n (−1, b1;q)n

and

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(3.10)

n

X

k=0

(q;q)_k(−1;q)_n · (−a₂)^k(b₂/a₂;q)_k

(b₂;q)_k = (a₂;q)_n (−1, b₂;q)_n. Substituting (3.8), (3.9) and (3.10) into (3.7), we obtain (1.1).

Takinga₂ = 1in (1.1), we get the following corollary.

Corollary 3.2. We have (3.11)

²

φ₁

b₁/a₁, q⁻ⁿ

b₁ ;q,−a₁qⁿ

≤λ(−1;q)_n.

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4. Some Applications of the Inequality

Convergence ofq-series is an important problem which is at times very difficult. As applications of the inequality derived in this paper, we obtain some results about the convergence of the q-series involving ₃φ₂. In this section, we mainly discuss the convergence of the followingq-series:

(4.1)

∞

X

n=0

c_n3φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

b₁, b₂ ;q,−a₁a₂qⁿ

.

Theorem 4.1. Suppose |a_i| ≤ 1 and b_i < ^aⁱ₂⁺¹ for i = 1,2. Let {c_n} be a real sequence satisfying

n→∞lim

c_n+1 c_n

=p <1.

Then the series (4.1) is absolutely convergent.

Proof. It is obvious thatb_i <1fori= 1,2. Combining the following inequality (4.2)

³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ b1, b2

;q,−a₁a₂qⁿ

−

(a₁, a₂;q)_n (−1, b1, b2;q)n

≤ ³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

− (a₁, a₂;q)_n (−1, b₁, b₂;q)_n

with (2.1), shows that

(4.3) ³

φ₂

b1/a1, b2/a2, q⁻ⁿ

≤

(a1, a2;q)n

(−1, b₁, b₂;q)_n

+λµ(−1;q)_n. Since

|a_i| ≤1, b_i < a_i+ 1

2 , i= 1,2,

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which is equivalent to

|ai| ≤1, |a_i−b_i|

1−b_i <1, i= 1,2, then

(4.4) λ =µ= 1.

Substituting (4.4) into (4.3), we obtain (4.5)

3φ2

b₁/a₁, b₂/a₂, q⁻ⁿ

≤

(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

+ (−1;q)n. Multiplication of the two sides of (4.5) by|c_n|gives

(4.6)

cn3φ2

b₁/a₁, b₂/a₂, q⁻ⁿ

≤

c_n(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

+|cn(−1;q)n|. The ratio test shows that both

∞

X

n=0

c_n(a₁, a₂;q)_n (−1, b1, b2;q)n

and

∞

X

n=0

c_n(−1;q)_n

are absolutely convergent. From (4.6), we get that (4.1) is absolutely convergent.

Theorem 4.2. Suppose|a₁| > 1ora₁ <2b₁ −1, b₁ < 1,|a₂| ≤ 1andb₂ ≤ ^a²₂⁺¹. Let{c_n}be a real sequence satisfying

n→∞lim

c_n+1 c_n

=p < 1 M, whereM = maxn

|a₁|,^|a_1−b¹^−b¹^|

1

o

. Then the series (4.1) is absolutely convergent.

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Proof. First we point out thata₁ <2b₁−1implies

|a₁ −b₁| 1−b₁ >1.

So, under the conditions of the theorem, we know λ=Mⁿ and µ= 1.

Multiplying both sides of (4.3) by|c_n|, one gets (4.7)

c_n3φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

≤

c_n(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

+|c_nMⁿ(−1;q)_n|. The ratio test shows that both

∞

X

n=0

cn(a1, a2;q)n

(−1, b₁, b₂;q)_n and

∞

X

n=0

c_nMⁿ(−1;q)_n

are absolutely convergent. From (4.7), we get that (4.1) is absolutely convergent.

Similarly, we have

Theorem 4.3. Suppose|a_i|>1ora_i <2b_i−1,b_i <1withi= 1,2. Let{c_n}be a real sequence satisfying

n→∞lim

c_n+1 c_n

=p < 1 M N, whereM = maxn

|a₁|,^|a_1−b¹^−b¹^|

1

o

andN = maxn

|a₂|,^|a_1−b²^−b²^|

2

o

. Then the series (4.1) is absolutely convergent.

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References

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VUORINEN, Inequalities for zero-balanced hypergeometric functions, Trans- actions of the American Mathematical Society, 347(5), 1995.

[2] G.E. ANDREWS, The Theory of Partitions, Encyclopedia of Mathematics and Applications; Volume 2. Addison-Wesley Publishers, 1976.

[3] W.N. BAILEY, Generalized hypergeometric series, Cambridge Math. Tract No.32, Cambridge University Press, London and New York.1960.

[4] G. GASPER AND M. RAHMAN, Basic Hypergeometric Series, Cambridge Univ.Press, Cambridge, MA, 1990.

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[9] MINGJIN WANG, An inequality for r+1φ_r and its applications, Journal of Mathematical Inequalities, 1(2007) 339–345.

[10] MINGJIN WANG, Two inequalities for _rφ_r and applications, Journal of In- equalities and Applications, 2008, Article ID 471527.