In this paper, we establish the log convexity and Tur´an type inequalities of extended (p, q)-beta functions

https://doi.org/10.5831/HMJ.2019.41.4.745

INEQUALITIES OF EXTENDED (p, q)-BETA AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

Shahid Mubeen, Kottakkaran Sooppy Nisar^∗, Gauhar Rahman, and Muhammad Arshad

Abstract. In this paper, we establish the log convexity and Tur´an type inequalities of extended (p, q)-beta functions. Likewise, we present the log-convexity, the monotonicity and Tur´an type inequal- ities for extended (p, q)-confluent hypergeometric function by uti- lizing the inequalities of extended (p, q)-beta functions.

1. Introduction

The Euler’s Beta function and its extensions together with special functions found in many recent papers( see, for details, [4, 5, 7]). In [7], (p, q) extension of Beta and hypergeometric functions (HFs) are defined by

(1) Bp,q(y, z) =

1

Z

0

t^y−1(1 − t)^z−1e^{−pt−1−tq} d t,

(min{<(y), <(z)} > 0; min{<(p), <(q)} > 0),

(2) Fp,q(d, e; c; x) =X

n≥0

(d)n

B(e + n, c − e; p, q) B(e, c − e)

xⁿ n!

(|x| < 1; <(c) > <(e) > 0)

Received February 9, 2019. Revised June 19, 2019. Accepted June 27, 2019.

2010 Mathematics Subject Classification. 33B15, 33B99.

Key words and phrases. Beta function, Extended hypergeometric functions, Tur´an-type inequalities, log-convexity.

*Corresponding author

and (3) Φ_p,q

e; c; x

=

∞

X

n=0

B(e + n; c − e; p, q) B(e, c − e)

xⁿ

n! (<(c) > <(e) > 0) Clearly, the particular cases of (1), (2) and (3) when p = q turns to p-extended Beta, p-extended Gauss HF and p-extended confluent HF of [6] as:

(4) B(y, z; p) =

1

Z

0

t^y−1(1 − t)^z−1e⁻

p t(1−t)d t

(<(p) > 0, <(y) > 0, <(z) > 0),

(5) Fp(d, e; c; z) =X

n≥0

(d)n

B(e + n, c − e; p) B(e, c − e)

zⁿ n!

(p ≥ 0, |z| < 1; <(c) > <(e) > 0) and

(6) Φp

 e; c; z



=

∞

X

n=0

B(e + n; c − e; p) B(e, c − e)

zⁿ n!

(p ≥ 0, <(c) > <(e) > 0) respectively. It is obvious, p = q = 0 in (1), (2) and (3) reduces to the classical Beta, Gauss HF and confluent HF, respectively. Here we need the following integral expression of (p, q)-extended confluent HF [7]:

Φp,q

 e; c; z



= 1

B(e, c − e) Z 1

0

t^e−1(1 − t)^c−e−1exp

 zt −p

t − q 1 − t

 dt, (7)

min{<(p), <(q)} ≥ 0, <(c) > <(e) > 0.

In this paper, we present the log-convexity, monotonicity and various other properties of the (p, q)-extended Beta function by using the clas- sical Holder-Rogers inequality and Chebyshev’s inequality for integrals.

Also other properties of (p, q)-extended HF are derived with the help of inequalities of (p, q)-extended beta functions. As the particular cases, we obtain Tu´ran-type inequalities for (p, q)-extended beta, (p, q)-extended confluent and (p, q)-extended Gaussian HFs.

2. Main results: Inequalities of extended (p, q)-beta function This section devoted to establish some inequalities involving (1) by using some natural inequalities [11]. We recall the following well-known Chebychev’s and H¨older-Rogers inequalities.

Lemma 2.1. (see [8, 9]) Suppose that the functions f, g : [a, b] ⊆ R → R are asynchronous ∀x ∈ [a, b] and r(x) : [a, b] ⊆ R → R is a function of positive integrable, then

b

Z

a

r(x)f (x)dx

b

Z

a

r(x)g(x)dx ≤

b

Z

a

r(x)dx

b

Z

a

r(x)f (x)g(x)dx.

(8)

Definition 2.2. A function g : (a, b) → R is said to be convex if for any y₁, y₂∈ (a, b) and β ∈ (0, 1),

g(βy₁+ (1 − β)y₂) ≤ βg(y₁) + (1 − β)g(y₂).

(9)

Which means that when we go from y₁ to y₂, the line joining the points (y1, g(y1)) and (y2, g(y2)) lies always above the graph of g.

Definition 2.3. A function g is known to be a log-convex if g > 0 and log g is convex i.e., for all y1, y2 ∈ I (where I is an interval) and β ∈ (0, 1), we have

log g(βy1+ (1 − β)y2) ≤ β log g(y1) + (1 − β) log g(y2)

= log(g^β(y)g^1−β(y2)).

This implies that

g(βy₁+ (1 − β)y₂) ≤ g^β(y₁)g^1−β(y₂).

(10)

Lemma 2.4. (H¨older inequality [13]) If θ₁ and θ₂ are positive real numbers such that _θ¹

1 + _θ¹

2 = 1, then the following inequality holds for integrable functions g, h : [a, b] → R:

|

b

Z

a

g(x)h(x)dx| ≤ (

b

Z

a

|g|^θ1dx)

1 θ1(

b

Z

a

|h|^θ2dx)

1 θ2. (11)

Now, we prove the following Theorems:

Theorem 2.5. If the positive real numbers x, y, x1, y1 satisfying the condition

(x − x₁)(y − y₁) ≥ 0, (12)

then for (1), we have the inequality

B_p,q(x, y₁)B_p,q(x₁, y) ≤ B_p,q(x₁, y₁)B_p,q(x, y), (13)

Proof. Let f, g, h : [0, 1] → [0, ∞) be given by f (t) = t^x−x1, g(t) = (1 − t)^y−y1 and

h(t) = t^x1−1(1 − t)^y1−1exp

−p t − q

1 − t

 . The derivative of f and g are

f⁰(t) = (x − x1)t^x−x1−1, g⁰(t) = (y1− y)(1 − t)^y−y1−1. This show that the monotonicity of f and g on [0, 1] are same . Employing (8) on the above functions f , g and h, we have

Z b a

t^x−1(1 − t)^y1−1exp



−p t − q

1 − t

 dt



Z b a

t^x1−1(1 − t)^y−1exp



−p t − q

1 − t

 dt



≤Z b a

t^x1−1(1 − t)^y1−1exp



−p t − q

1 − t

 dt



Z b a

t^x−1(1 − t)^y−1exp



−p t − q

1 − t

 dt

 , which implies that,

B_p,q(x, y₁)B_p,q(x₁, y) ≤ B_p,q(x₁, y₁)B_p,q(x, y), which is the desired proof.

Theorem 2.6. The function (p, q) 7→ B_p,q(z₁, z₂) is log convex on (0, ∞) for each z₁, z₂ > 0. Moreover, the function B_p,q(z₁, z₂) satisfy the following Tur´an type inequality

B_p,q² (z1, z2) − Bp+a,q+a(z1, z2)Bp−a,q−a(z1, z2) ≤ 0, (14)

for all real a.

Proof. In view of log-convexity definition, it will be sufficient to prove that

(15) B_{αp1+(1−α)p2,αq1+(1−α)q2}(z₁, z₂)

≤

Bp1,q1(z1, z2)

α

Bp2,q2(z1, z2)

1−α

,

for α ∈ [0, 1], p1, p2, q1, q2 > 0 and for a fixed z1, z2> 0. Obviously, (15) is true for α = 0 and α = 1. Assume that α ∈ (0, 1), then it follows from (1) that

B_{αp1+(1−α)p2,αq1+(1−α)q2}(z1, z2)

= Z 1

0

t^z1−1(1 − t)^z2−1exp−αp₁− (1 − α)p₂

t +−αq₁− (1 − α)q₂ 1 − t

 dt

=

Z 1 0

t^z1−1(1 − t)^z2−1exp



−p₁ t − q₁

1 − t

 dt

α

×Z 1 0

t^z1−1(1 − t)^z2−1exp

−p₂ t − q₂

1 − t

 dt1−α

(16)

Let θ1 = ¹_α and θ2 = _(1−α)¹ . Clearly θ1 > 1 and θ1 + θ2 = θ1θ2. Thus applying the H¨older-Rogers inequality (11) for integrals in (16) gives

B_{αp1+(1−α)p2,αq1+(1−α)q2}(z1, z2) (17)

<

Z 1 0

t^z1−1(1 − t)^z2−1exp



−p1

t − q1

1 − t

 dt

α

×Z 1 0

t^z1−1(1 − t)^z2−1exp

−p₂ t − q₂

1 − t

 dt1−α

=

B_p1,q1(z₁, z₂)α

B_p2,q2(z₁, z₂)1−α

, (18)

This implies that (p, q) 7→ B_p,q(z₁, z₂) is log convex on (0, ∞).

Now, taking α = ¹₂, p1 = p − a, p2 = p + a, and q1 = q − a, q2 = q + a, the inequality (18) yields

B_p,q² (z₁, z₂) − B_p+a,q+a(z₁, z₂)B_p−a,q−a(z₁, z₂) ≤ 0.

Theorem 2.7. The function (y, z) 7→ Bp,q(y, z) is logarithmic convex on (0, ∞) × (0, ∞), for all p, q ≥ 0. Especially,

B_p,q² y₁+ y₂

2 ,z₁+ z₂ 2

≤ B_p,q(y₁, z₁)B_p,q(y₂, z₂).

Proof. Let (y₁, z₁), (y₂, z₂) ∈ (0, ∞)², and c, d ≥ 0 with c + d = 1, then we have

Bp,q



c(y1, z1) + d(y2, z2)



= Bp,q(cy1+ dy2, cz1+ dz2).

(19)

Applying the definition of (p, q)-extended beta function on the right hand side of (19), we have

Bp,q



c(y1, z1) + d(y2, z2)



=

1

Z

0

t^cy1+dy2−1(1 − t)^cz1+dz2−1exp

−p t − q

1 − t

 dt

=

1

Z

0

t^{cy1+dy2−(c+d)}(1 − t)^{cz1+dz2−(c+d)}exp



−p(c + d)

t −q(c + d) 1 − t

 dt

=

1

Z

0

t^c(y1−1)t^d(y2−1(1 − t)^c(z1−1)(1 − t)^d(z2−1)

exp

− pc t − qc

1 − t

 exp

−pd t − qd

1 − t

 dt

= Z1

0



t^y1−1(1 − t)^z1−1exp

−p t − q

1 − t

c



t^y2−1t^z2−1exp



−p t − q

1 − t

d

dt.

Again by considering θ₁ = ¹_c, θ₂ = ¹_d, we can use the H¨older-Rogers inequality for above integrals and it follows

B_p,q

c(y₁, z₁) + d(y₂, z₂)

≤

1

Z

0

t^y1−1(1 − t)^z1−1exp



−p t − q

1 − t

 dt

c

×

1

Z

0

t^y2−1t^z2−1exp

−p t − q

1 − t

 dtd

=

B_p,q(y₁, z₁)c

B_p,q(y₂, z₂)d

.

This shows the logarithmic convexity of extended (p, q)-beta function B_p,q(y, z) on (0, ∞)².

For c = d = ¹₂, we have B_p,q² y₁+ y₂

2 ,z₁+ z₂ 2

≤ B_p,q(y₁, z₁)B_p,q(y₂, z₂).

(20)

Let y, z > 0 be such that mina∈R(y + a, y − a) > 0, then by taking y₁ = y + a, y₂= y + a, z₁ = z + b and z₂ = z − b in (20), we get

h

B_p,q(y, z)i2

≤ B_p,q(y + a, z + b)B_p,q(y − a, z − b), (21)

for all p, q ≥ 0.

3. Inequalities for (p, q)-extended confluent hypergeometric function

This section presents the log-convexity and Tur´an type inequality for extended confluent hypergeometric function defined in (3). We recall the following:

Lemma 3.1. [2] Let f (x) = P

n≥0anxⁿ and g(x) = P

n≥0bnxⁿ are the power series, where an ∈ R and b_n > 0, ∀n. Further assume that both series converge on |x| < α. If the sequence {a_n/b_n}_n ≥ 0 is increasing (or decreasing), then x 7→ f (x)/g(x) is also increasing (or decreasing) function on (0, α).

This lemma is valid only if both f and g are both even or both odd functions.

Theorem 3.2. Let µ ≥ 0 and η, ξ > 0, then the following assertions for extended (p, q)-confluent hypergeometric function are true.

(i) For η ≥ ξ, the function x 7→ Φ_p,q

µ; η; x /Φ_p,q

µ; ξ; x

is increasing on (0, ∞).

(ii) For η ≥ ξ, we have ξΦ_p,q

µ + 1; η + 1; x Φ_p,q

µ; ξ; x

≥ ηΦ_p,q(µ; η; x)Φ_p,q

µ + 1; ξ + 1; x . (iii) The function x 7→ Φ_p,q

µ; η; x

is log-convex on R.

(iv) The function (p, q) 7→ Φ_p,q

µ; η; x

is log convex on (0, ∞) for fixed x > 0.

(v) Let σ > 0. then the function

µ 7→

B(µ, η)Φ_p,q

µ + σ; η; x B(µ + σ, η)Φ_p,q

µ; η; x is decreasing on (0, ∞) for fixed η, x > 0.

Proof. From the definition of (3), it follows that

Φ_p,q

µ; η; x Φ_p,q

µ; ξ; x = P∞

n=0a_n(c)xⁿ P∞

n=0a_n(d)xⁿ, where a_n(t) = B_p,q(µ + n, t − µ) B(µ, t − µ)n! . (22)

If we denote f_n= a_n(c)/a_n(d), then fn− f_n+1=an(c)

a_n(d)− an+1(c) a_n+1(d)

=B(µ, ξ − µ) B(µ, η − µ)

Bp,q(µ + n, η − µ)

B_p,q(µ + n, η − µ) −Bp,q(µ + n + 1, η − µ) B_p,q(µ + n + 1, η − µ)

 . Now take x = µ + n, y = η − µ, x₁ = µ + n + 1, y₁ = η − µ in (13). Since (x − x1)(y − y1) = η − ξ ≥ 0, it follows from Theorem 2.5 that

B_p,q(µ + n, η − µ)

Bp,q(µ + n, ξ − µ) ≤ B_p,q(µ + n + 1, η − µ) Bp,q(µ + n + 1, ξ − µ),

this means that {fn} is an increasing sequence and using Lemma 3.1, we observe that x 7→ Φ_p,q

µ; η; x /Φ_p,q

µ; ξ; x

is increasing on (0, ∞).

To prove the assertion (ii), we need the identity from [7]:

dⁿ dxⁿΦ_p,q

µ; η; x

= (µ)_n (η)n

Φ_p,q

µ + n; η + n; x . (23)

Since the increasing properties of x 7→ Φp,q

 µ; η; x

 /Φp,q

 µ, ξ; x

 is same as the following inequality

d dx

 Φp,q



µ; η; x Φ_p,q

µ, ξ; x

≥ 0.

(24)

This together with (23) implies Φ⁰_p,q

µ; η; x Φ_p,q

µ; ξ; x

− Φ_p,q

µ; η; x Φ⁰_p,q

µ; ξ; x

= µ

ηΦ_p,q

µ + 1; η + 1; x

Φ_p,q(µ; ξ; x)

− µ

ξΦp,q

 µ; η; x

 Φp,q



µ + 1; ξ + 1; x



≥ 0.

This implies that ξΦp,q



µ + 1; η + 1; x



Φp,q(µ; ξ; x) ≥ ηΦp,q

 µ; η; x

 Φp,q



µ + 1; ξ + 1; x



which prove the assertion.

Using (7), we can prove the log-convexity of x 7→ Φ_p,q

β; γ; x and by applying the H¨older-Rogers inequality for integrals, we get

Φ_p,q

µ; η; αx + (1 − α)y

= 1

B(µ, η −µ) Z 1

0

t^µ−1(1−t)^η−µ−1exp

αxt + (1−α)yt − p t − q

1 − t

 dt

= 1

B(µ, η − µ) Z 1

0

h

t^µ−1(1 − t)^η−µ−1exp xt −p

t − q 1 − t

α

×

t^µ−1(1 − t)^η−µ−1exp

 yt −p

t − q 1 − t

1−αi dt

≤h 1

B(µ, η − µ) Z ₁

0

t^µ−1(1 − t)^η−µ−1exp xt −p

t − q 1 − t

 dtiα

×h 1

B(µ, η − µ) Z 1

0

t^µ−1(1 − t)^η−µ−1exp

 xt −p

t − q 1 − t

 dt

i1−α

= Φ_p,q

µ; η; xα Φ_p,q

µ; η; y1−α

, (x, y > 0, α ∈ [0, 1]).

This prove that x 7→ Φ_p,q

µ; η; x

is log-convex for a fixed x > 0. For the case when x < 0, then the assertion immediately follows from the identity (see [7]):

Φ_p,q

µ; η; x

= e^xΦ_q,p

η − µ; η; −z .

Since, the infinite sum of log-convex functions is log-convex for x > 0.

Thus, the log-convexity of (p, q) 7→ Φ_p,q

µ; η; x

is equivalent to prove that (p, q) 7→ B(µ+n, η −µ) is log-convex on (0, ∞) and for non-negative integer n. From Theorem 2.6, it is clear that (p, q) 7→ B(µ + n, η − µ) is log-convex for η > µ > 0 and hence assertion (iv) is true.

Now, let µ⁰ ≥ µ and set h(t) = t^µ0−1(1 − t)^η−µ0−1exp



xt − ^p_t −

q 1−t



, f (t) = 

t 1−t

µ−µ⁰

and g(t) = 

t 1−t

σ

. Then using the integral representation (7) of extended confluent hypergeometric function, we

have

B(µ, η)Φp,q



µ + σ; η; x

 B(µ + σ, c)Φp,q

 µ; η; x

 −

B(µ⁰, η)Φp,q



µ⁰+ σ; η; x

 B(µ⁰+ σ, η)Φp,q

 µ⁰; η; x



= R₁

0 f (t)g(t)h(t)dt R1

0 f (t)h(t)dt − R₁

0 g(t)h(t)dt R1

0 h(t)dt . (25)

One can easily determine that for µ⁰ ≥ µ, the function f is decreasing when σ ≥ 0 and the function g is increasing . Since h is non negative function for t ∈ [0, 1]. Thus, by reverse Chebyshev’s reverse inequality (8), it follows that

Z 1 0

f (t)h(t)dt Z 1

0

g(t)h(t)dt ≤ Z 1

0

h(t)dt Z 1

0

f (t)g(t)h(t)dt.

(26)

This together with (25) implies B(µ, η)Φ_p,q

µ + σ; η; x B(µ + ξ, η)Φ_p,q

µ; η; x −

B(µ⁰, η)Φ_p,q

µ⁰+ σ; η; x B(µ⁰+ σ, η)Φ_p,q

µ⁰; η; x ≥ 0, which is same as

β 7→

B(µ, η)Φ_p,q

µ + σ; η; x B(µ + σ, η)Φ_p,q

µ; η; x is decreasing on (0, ∞).

Remark 3.3. In particular, the following decreasing property of ex- tended (p, q)-confluent hypergeometric function

β 7→

B(µ, η)Φp,q



µ + σ; η; x

 B(µ + σ, η)Φp,q

 µ; η; x

 is equivalent to the following inequality

Φ²_p,q

µ + σ; η; x

≥ B²(µ + σ, η)

B(µ + 2σ, η)B(µ, η)Φ_p,q

µ + 2σ; η; x Φ_p,q

µ; η; x . (27)

When p = q, then the above inequality will reduce to the inequality recently proved by [12]. Similarly, when p = q = 0, then the above inequality reduces to the inequality of confluent hypergeometric which is an improved version of Theorem 4(b) given in [10].

4. Conclusion

In this paper, we introduced inequalities for extended (p, q)-beta and (p, q)-confluent hypergeometric function defined by Choi et al. [7].

Throughout in this paper, if we take p = q then we get the inequalities of extended beta function and extended confluent hypergeometric func- tion recently introduced by Mondal [12]. Similarly if we take p = q = 0, then the newly defined inequalities for extended (p, q)-beta function will reduce to the inequalities of classical beta function (see [1, 8]).

Declaration

The pre-print version of this article available at ‘https://arxiv.org/abs/1703.

08852’.

References

[1] R. P. Agarwal, N. Elezovi´c and J. Peˇcari´c, On some inequalities for beta and gamma functions via some classical inequalities, J. Inequal. Appl., 2005(5)(2005), 593-613.

[2] M. Biernacki and J. Krzyz, On the monotonicity of certain fractionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sklodowska. Sect. A., 9 (1955), 135-147.

[3] S. Butt, J. Pecaric and A. Rehman, Exponential convexity of Petrovic and related functional, J. Inequal. Appl., 2011(2011), 89.

[4] M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-123.

[5] M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math., 78 (1997), 19-32.

[6] M.A. Chaudhry, A. Qadir, H.M. Srivastava and R.B. Paris, Extended hyper- geometric and confluent hypergeometric functions, Appl. Math. Comput. , 159 (2004), 589-602.

[7] J. Choi, A. K. Rathie and R. K. Parmar, Extension of extended beta, hyper- geometric and confluent hypergeometric functions, Honam Mathematical J., 36 (2014), 357-385.

[8] S. S. Dragomir, R. P. Agarwal and N. S. Barnett, Inequalities for beta and gamma functions via some classical and new inequalities, J. Inequal. Appl., 5(2) (2000), 103-165.

[9] P. Kumar, S. P. Singh and S. S. Dragomir, Some inequalities involving beta and gamma functions, Nonlinear Anal. Forum, 6(1)(2001), 143-150.

[10] D. Karp and S. M. Sitnik, Log-convexity and log-concavity of hypergeometric-like function, J. Math. Anal. Appl., 364(2) (2010), 384-394.

[11] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht (1993).

[12] S. R. Mondal, Inequalities of extended beta and extended hypergeometric func- tions, J. Inequal. Appl., (2017) 2017,10

[13] W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill International Edi- tions (1987).

[14] P. Tur´an, On the zeros of the polynomials of Legendre, Casopis pro Pestovani Mat. a Fys, 75 (1950), 113-122.

Shahid Mubeen

Department of Mathematics, University of Sargodha, Sargodha, Pakistan

E-mail:[email protected] Kottakkaran Sooppy Nisar

Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, 11991,

Prince Sattam Bin Abdulaziz University, Kingdom of Saudi Arabia E-mail:[email protected], [email protected]

Gauhar Rahman

Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Khyber Pakhtoonkhwa, Pakistan

E-mail: [email protected] Muhammad Arshad

Department of Mathematics, International Islamic University, Islamabad, Pakistan

E-mail: marshad [email protected]