New properties for several classes of functions related to the Fox–Wright functions

Journal of Computational and Applied Mathematics 362 (2019) 161–171

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New properties for several classes of functions related to the Fox–Wright functions Khaled Mehrez

∗

Département de Mathématiques, Issat Kasserine, Université de Kairouan, Tunisia Département de Mathématiques, Faculté des sciences de Tunis, Université Tunis El Manar, Tunisia

article

info

Article history: Received 2 March 2019 Received in revised form 15 May 2019 MSC: 11M35 33D05 33B15 26A51

a b s t r a c t In this paper our aim is to establish several new properties for some class of functions related to the Fox–Wright functions. In particular, the monotonicity, convexity, subadditivity and super-additivity properties involving the Fox–Wright functions are proved, These results are also closely connected with some functional inequalities (such as Turán-type inequalities). Moreover, we investigate certain criteria for the univalence and starlikeness for a certain class of functions associated with the Fox–Wright functions. © 2019 Elsevier B.V. All rights reserved.

Keywords: Fox–Wright function Turán-type inequalities Starlike functions Univalent functions

1. Introduction and preliminaries We will use p Ψq [.] to denote the Fox–Wright generalization of the familiar hypergeometric p Fq function with p numerator and q denominator parameters, defined by [1, p. 4, Eq. (2.4)] p Ψq

[

(a1 ,A1 ),...,(ap ,Ap ) (b1 ,B1 ),...,(bq ,Bq )

∞ ∏p ⏐ ] ⏐ ] ∑ [ Γ (αl + kAl ) z k (ap ,Ap ) ⏐ ⏐ ∏lq=1 z , = Ψ z = ⏐ p q (bq ,Bq ) ⏐ k! l=1 Γ (βl + kBl ) k=0

(1.1)

where, (Al ≥ 0, l = 1, . . . , p; Bl ≥ 0, and l = 1, . . . , q). The convergence conditions and convergence radius of the series at the right-hand side of (1.1) immediately follow from the known asymptotics of the Euler Gamma-function. The defining series in (1.1) converges in the whole complex z-plane when

∆=

q ∑ j=1

Bj −

p ∑

Ai > −1.

i=1

∗ Correspondence to: Département de Mathématiques, Faculté des sciences de Tunis, Université Tunis El Manar, Tunisia. E-mail address: [email protected]. https://doi.org/10.1016/j.cam.2019.05.025 0377-0427/© 2019 Elsevier B.V. All rights reserved.

(1.2)

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K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

If ∆ = −1, then the series in (1.1) converges for |z | < ρ , and |z | = ρ under the condition ℜ(µ) > where

( ρ=

p ∏

⎞ )⎛ q q p ∏ ∑ ∑ p−q Bj −A Ai i ⎝ Bj ⎠ , µ = bj − ak +

i=1

j=1

j=1

2

k=1

1 , 2

(see [2] for details),

(1.3)

The generalized hypergeometric function p Fq is defined by p Fq

[

a1 ,...,ap b1 ,...,bq

∞ ∏p ⏐ ] ∑ (al )k z k ⏐ ∏ql=1 ⏐z = k! l=1 (bl )k

(1.4)

k=0

where, as usual, we make use of the following notation: (τ )0 = 1, and (τ )k = τ (τ + 1)...(τ + k − 1) =

Γ (τ + k) , k ∈ N, Γ (τ )

to denote the shifted factorial or the Pochhammer symbol. Obviously, we find from the definitions (1.1) and (1.4) that (a1 ,1),...,(ap ,1) p Ψq (b1 ,1),...,(bq ,1)

[

⏐ ] ⏐ ] Γ (a )...Γ (a ) [ a1 ,...,ap ⏐ ⏐ 1 p ⏐z = p Fq b1 ,...,bq ⏐z . Γ (b1 )...Γ (bq )

(1.5)

For some interesting new properties involving the further properties of the generalized hypergeometric functions and generating functions associated with them and different extensions of various hypergeometric operators of fractional derivatives. A detailed account of such operators along with their properties and applications has been considered by several authors [3–10] and the references therein. The H-function was introduced by Fox in [11] as a generalized hypergeometric function defined by an integral representation in terms of the Mellin–Barnes contour integral Hqm,p,n

) ) ( ⏐ ( ⏐ ⏐(B1 ,β1 ),...,(Bq ,βq ) ⏐(Bq ,βq ) = Hqm,p,n z ⏐ z⏐ (A1 ,α1 ),...,(Ap ,αp ) (Ap ,αp ) ∏m ∏n ∫ 1 j=1 Γ (Aj s + αj ) j=1 Γ (1 − βj − Bj s) ∏q ∏p = z −s ds. 2iπ L j=n+1 Γ (Bk s + βk ) j=m+1 Γ (1 − αj − Aj s)

(1.6)

Here L is a suitable contour in C and z −s = exp(−s log |z | + i arg(z)), where log |z | represents the natural logarithm of |z | and arg(z) is not necessarily the principal value. The definition of the H-function is still valid when the Ai ’s and Bj ’s are positive rational numbers. Therefore, the Hfunction contains, as special cases, all of the functions which are expressible in terms of the G-function. More importantly, it contains the Fox–Wright generalized hypergeometric function defined in (1.1), the generalized Mittag-Leffler functions, etc. For example, the function p Ψq [.] is one of these special case of H-function. By the definition (1.1) it is easily extended to the complex plane as follows [12, Eq. 1.31], p Ψq

(αp ,Ap ) (βq ,Bq )

[

( ⏐ ⏐ ] ⏐ ⏐(Ap ,1−αp ) 1 ,q ⏐z = Hp,q+1 −z ⏐

(0,1),(Bq ,1−βq )

)

.

(1.7)

The representation (1.7) holds true only for positive values of the parameters Ai and Bj . In a series of recent papers published by many authors including Mehrez alone and/or with his co-workers Sitnik, Tomovski [13], Srivastava [14] and Baricz [15] have studied certain functional inequalities and geometric properties for some special functions, for example, the classical Gauss and Kummer hypergeometric functions, as well as the generalized hypergeometric functions [16–18], classical and generalized Mittag-Leffler functions [19], the Wright function [20], the Fox–Wright function [21], the Mathieu-type series and Volterra functions [22,23]. This paper is a continuation of some of the author’s previous results. Here, in our present investigation, our aim is to show some mean Turán-type inequalities and sub-additivity (super-additivity) properties associated with the Fox–Wright functions. In addition, we give a set of sufficient conditions for some classes of functions involving to the Fox–Wright function to be univalent and starlike. This paper is organized as follows: In Section 2, we show some Turán-type inequalities for the Fox–Wright function. In addition, we establish the monotonicity of ratios for the Fox–Wright functions, the results are also closely connected with Turán-type inequalities. In Section 3, we derive the sub-additivity and super-additivity properties of a class of functions related to the Fox–Wright functions. The main purpose, in Section 4, is to investigate certain criteria for the univalence and starlikeness for the Fox–Wright function p+1 Ψp [.]. 2. Turán-type inequalities for the Fox–Wright functions Our first main result is asserted by the following Theorem.

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

163

Theorem 1. Let a, b, σ , A be a positive real numbers such that b > a. The following integral representation 2 Ψ1

[

(σ ,1),(a,A) (b,A)

⏐ ] ⏐ ⏐z =

1

∫

1 AΓ (b − a)

a

1

t A −1 (1 − t A )b−a−1

dt .

(1 − tz)σ

0

(2.1)

holds true for all z ∈ C such that |z | < 1.. Furthermore, the function a ↦ → Γ (a).2 Ψ1

[

(a,1),(b−a,A) (b,A)

⏐ ] ⏐ ⏐z ,

is log-convex on (0, b). Moreover the following Turán-type inequality 2 Ψ1

[

(a+2,1),(b−a−2,A) (b,A)

⏐ ] ⏐ ] [ ⏐ (a,1),(b−a,A) ⏐ ⏐z 2 Ψ1 (b,A) ⏐z −

a

(

a+1

2 Ψ1

[

(a,1),(b−a,A) (b,A)

⏐ ])2 ⏐ ≥ 0, ⏐z

(2.2)

holds true for all 0 < z < 1. Proof. In [21, Theorem 4], we proved that the Fox–Wright function possesses the following integral representation p+1 Ψp

(σ ,1),(αp ,A) (βp ,A)

[

⏐ ] ∫ ⏐ ⏐z =

dµ(t)

1

(1 − tz)σ

0

,

(2.3)

where dµ(t) = Hpp,,p0

) (⏐ ⏐(A,bp ) dt t⏐ ,

(2.4)

t

(A,ap )

p,0

and Hp,p [.] is the Fox H-function. In particular, we get 2 Ψ1

[

(σ ,1),(a,A) (b,A)

⏐ ] ∫ ⏐ ⏐z =

1

t −1 (1 − tz)σ

0

1,0

H1,1

(⏐ ) ⏐(A,b) dt . t⏐

(2.5)

(A,a)

By using the identity 1,0

H1,1

(⏐ ) a 1 t A (1 − t A )b−a−1 ⏐(A,b) = , A > 0, b > a > 0, t⏐ (A,a) AΓ (b − a)

and (2.5), we get 2 Ψ1

[

(σ ,1),(a,A) (b,A)

⏐ ] ⏐ ⏐z =

1

∫

1 AΓ (b − a)

a

1

t A −1 (1 − t A )b−a−1

dt .

(1 − tz)σ

0

(2.6)

This implies that the following integral representation

Ωa [b, A; z ] := Γ (a).2 Ψ1

[

(a,1),(b−a,A) (b,A)

⏐ ] 1∫ ⏐ ⏐z = A

1

]a

1

[

1−tA 1

t A (1 − tz)

0

b

t A −1 1

(1 − t A )

dt .

(2.7)

holds true for all 0 < a < b, A > 0 and z ∈ (0, 1). Let us recall the Rogers–Hölder–Riesz inequality [24, p. 54], that is b

∫

b

[∫

|f (t)|p dt

|f (t)g(t)|dt ≤ a

] 1p [∫

a

b

|g(t)|p dt

] 1q

,

(2.8)

a

where p ≥ 1, 1p + 1q = 1, f and g are real functions defined on (a, b) and |f |p , |g |q are integrable functions on (a, b). From (2.7) and (2.8), for a1 , a2 ∈ (0, b), z ∈ (0, 1) and α ∈ [0, 1], we obtain

Ωαa1 +(1−α)a2 [b, A; z ] =

1

∫

{[

1

1−tA

]a1

1

{∫

1

≤

[

1

1−tA 1

0

At A (1 − tz)

}α {[

1

At A (1 − tz)

0

b

t A −1

}α {∫

1

1

dt

(1 − t A )

[

dt

(1 − t A ) 1

1−tA 1

0

}1−α

b

t A −1 1

At A (1 − tz)

b

t A −1

]a2

1

(1 − t A )

]a1

1

1−tA

At A (1 − tz)

]a2

}1−α

b

t A −1 1

(2.9)

dt

(1 − t A )

= [Ωa1 [b, A; z ]]α .[Ωa2 [b, A; z ]]1−α , this implies that the function a ↦ → Ωa [b, A; z ] is log-convex on (0, b), and hence for all a1 , a2 ∈ (0, b), 0 < z < 1 and t ∈ [0, 1], we have

[ ]t [ ]1−t Ωta1 +(1−t)a2 [b, A; z ] ≤ Ωa1 [b, A; z ] Ωa2 [b, A; z ] . Choosing a1 = a, a2 = a + 2 and t = Theorem 1 is thus completed. □

1 2

the above inequality reduces to the Turán-type inequality (2.2). The proof of

164

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

Theorem 2. The following assertions are true: a. Let ai and bi be a real numbers such that bi ≥ ai for all 1 ≤ i ≤ p. Then the following Turán-type inequality holds true: (ap ,Ap ) p Ψp (bp ,Ap )

[

⏐ ] [ ⏐ ] ( [ ⏐ ])2 (ap +2,Ap ) ⏐ (ap +1,Ap ) ⏐ ⏐ ≥ 0, z > 0. ⏐z p Ψp (bp +2,Ap ) ⏐z − p Ψp (bp +1,Ap ) ⏐z

(2.10)

b. Let ai , bi and Ai be a positive real numbers, such that k ∑

(H) : 0 < a1 ≤ a2 ≤ · · · ≤ ap , 0 < b1 ≤ b2 ≤ · · · ≤ bp ,

bj −

k ∑

j=1

aj ≥ 0, k = 1, . . . , p.

j=1

(ap ,Ap ) (bp ,Ap )

⏐ ] ⏐ ⏐z is log-convex on (0, ∞). Furthermore, the following Turán-type inequality ⏐ ] [ ⏐ ] ( [ ⏐ ])2 [ (ap ,Ap ) ⏐ (ap +2Ap ,Ap ) ⏐ (ap +Ap ,Ap ) ⏐ ≥ 0, p Ψp (bp ,Ap ) ⏐z p Ψp (bp +2Ap ,Ap ) ⏐z − p Ψp (bp +Ap ,Ap ) ⏐z

Then, the function z ↦ → p Ψp

[

(2.11)

holds for all z > 0. Proof. a. Let us introduce the following notations

Ψn (ap , bp , Ap ; z) =

n ∑

Ωk (ap , bp , Ap )z k , where Ωk (ap , bp , Ap ) =

k=0

p ∏ Γ (ai + kAi ) , k ∈ {0, 1, . . . , n} . k!Γ (bi + kAi ) i=1

Using the Cauchy–Buniakowsky–Schwarz inequality [24, p. 41] one has

( Ψn (ap , bp , Ap ; z)Ψn (ap + 2, bp + 2, Ap ; z) =

n ∑

)( Ωk (ap , bp , Ap )

k=0

≥

[ n ∑√

n ∑

) Ωk (ap + 2, bp + 2, Ap )z

k

k=0

]2 Ωk (ap , bp , Ap )Ωk (ap + 2, bp + 2, Ap )z

k

.

k=0

In order to prove that

Ψn (ap , bp , Ap ; z)Ψn (ap + 2, bp + 2, Ap ; z) ≥ [Ψn (ap + 1, bp + 1, Ap ; z)]2 ,

(2.12)

holds, we just need to show that

Ωk (ap , bp , Ap )Ωk (ap + 2, bp + 2, Ap ) ≥ [Ωk (ap + 1, bp + 1, Ap )]2 ,

(2.13)

holds for all ∈ {0, 1, . . . , n}. Observe that (2.13) is equivalent to the inequality

[ ]2 Γ (ai + kAi )Γ (ai + kAi + 2) Γ (ai + kAi + 1) ≥ , Γ (bi + kAi )Γ (bi + kAi + 2) Γ (bi + kAi + 1) which is equivalent to ai + kAi + 1 bi + kAi + 1

≥

ai + kAi bi + kAi

, ⇐⇒ bi ≥ ai , for all 1 ≤ i ≤ p.

Thus, if n tends to infinity in (2.12), then we obtain the required inequality. b. From again the Rogers–Hölder–Riesz inequality (2.8) and using the following integral representation [21, Theorem 1] (ap ,Ap ) p Ψp (bp ,Ap )

[

⏐ ] ∫ ⏐ ⏐z =

1 0

[⏐ ] ⏐(Ap ,bp ) dt

ezt Hpp,,p0 t ⏐ p,0

[⏐ ] ⏐(Ap ,bp )

and using the fact that the Hp,p t ⏐ and λ ∈ [0, 1], p Ψp

[

(ap ,Ap ) (bp ,Ap )

t

(Ap ,ap )

(Ap ,ap )

⏐ ] ∫ ⏐ λ z + (1 − λ )z ⏐ 1 2 =

1

∫

1

,

(2.14)

is non-negative on (0, 1), (see [21, Remark 2]), we hind that for z1 , z2 ∈ (0, 1)

[⏐ ] ⏐(Ap ,bp ) dt

e(λz1 +(1−λ)z2 )t Hpp,,p0 t ⏐

t

(Ap ,ap )

0

[

=

e

z1 t

t

0

[⏐ ]]λ [ z t [⏐ ]]1−λ e 2 p,0 ⏐(Ap ,bp ) ⏐(Ap ,bp ) Hpp,,p0 t ⏐ Hp,p t ⏐ dt (Ap ,ap )

t

(Ap ,ap )

[⏐ ] ]λ [∫ 1 z t [⏐ ] ]1−λ e e 2 p,0 ⏐(Ap ,bp ) ⏐(Ap ,bp ) p,0 ≤ Hp,p t ⏐ dt Hp,p t ⏐ dt (Ap ,ap ) (Ap ,ap ) t t 0 0 ⏐ ]]1−λ ⏐ ]]λ [ [ [ [ (a ,A ) ⏐ (ap ,Ap ) ⏐ = p Ψp (bpp ,App ) ⏐z1 , p Ψp (bp ,Ap ) ⏐z2 [∫

1

z1 t

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

165

which readily implies that the function z ↦ → p Ψp

(ap ,Ap ) (bp ,Ap )

[

⏐ ] ⏐ ⏐z

is log-convex ⏐ ] on (0, ∞). Now, focusing on the Turán-type inequality (2.11), using the fact that the function z ↦→ [ (ap ,Ap ) p Ψp (bp ,Ap )

⏐ ⏐z is log-convex, we find

(ap ,Ap ) p Ψp (bp ,Ap )

[

⏐ ] (d ⏐ ] )2 ⏐ ] d2 [ [ (ap ,Ap ) ⏐ (ap ,Ap ) ⏐ ⏐ Ψ Ψ z − ≥ 0, ⏐ ⏐z p p p p (bp ,Ap ) (bp ,Ap ) ⏐z 2 dz

dz

and the derivative formula d dz

(ap ,Ap ) p Ψp (bp ,Ap )

[

⏐ ] ⏐ ] [ (ap +Ap ,Ap ) ⏐ ⏐ ⏐z = p Ψp (bp +Ap ,Ap ) ⏐z

yields the asserted result (2.11).

(2.15)

□

Remark 1. The idea of the proof of the Turán type inequality (2.10) is taken from [25]. For the proof of the next result we will need the following two lemmas. Let (an )n≥0 and ((bn )n≥0 be) two sequences of real numbers. If bn > 0 for n ≥ 0 and if the sequence (an /bn ) is

Lemma 1.

increasing (decreasing), then

a1 +···+an b1 +···+bn

is increasing (decreasing).

n ≥0

The next lemma is about the monotonicity of two power series. For more details, one may see [26]. Lemma 2. Let (an )n≥0 and (bn )n≥0 be two sequences of real numbers and let the power series f (x) =

∑

n ≥0

an xn and

g(x) = n≥0 bn x be convergent for all |z | < r. If bn > 0 for n ≥ 0 and if the sequence (an /bn ) is increasing (decreasing), then the function x ↦ → f (x)/g(x) is increasing (decreasing) on (0, r). n

∑

Let µ be a real number such that bi > max(Ai , µ) > 0 and let

Theorem 3.

Φ (ap , bp , Ap , µ, z) = p Ψp

[

(ap ,Ap ) (bp −µ,Ap )

⏐ ])2 ⏐ ]/ ( [ ⏐ ] [ (ap ,Ap ) ⏐ (ap ,Ap ) ⏐ ⏐ , z > 0, ⏐z p Ψp (bp +µ,Ap ) ⏐z p Ψp (bp ,Ap ) ⏐z

(2.16)

Then, the function z ↦ −→ Φ (ap , bp , Ap , µ, z) is increasing on (0, ∞). Moreover, the following Turán-type inequality (ap ,Ap ) p Ψp (bp −µ,Ap )

[

p ⏐ ] ∏ ⏐ ] [ (ap ,Ap ) ⏐ ⏐ ⏐z p Ψp (bp +µ,Ap ) ⏐z − i=1

⏐ ])2 ( [ Γ 2 (bi ) (ap ,Ap ) ⏐ ≥ 0, p Ψp (bp ,Ap ) ⏐z Γ (bi − µ)Γ (bi + µ)

(2.17)

holds true for all z > 0. Proof. By using the Cauchy product, we have that

∑∞ Ak z k Φ (ap , bp , Ap , µ, z) = ∑k∞=0 , k k=0

(2.18)

Bk z

where Ak =

p k ∑ ∏ j=0 i=1

k

p

∑ ∏ Γ (ai + jAi )Γ (ai + (k − j)Ai ) Γ (ai + jAi )Γ (ai + (k − j)Ai ) , and Bk = . Γ (bi − µ + jAi )Γ (bi + µ + (k − j)Ai ) Γ (bi + jAi )Γ (bi + (k − j)Ai ) j=0 i=1

Let the sequences (uk,j )j≥0 , (vk,j )j≥0 and (wk,j )j≥0 be defined, respectively, by p

uk,j =

p ∏ i=1

p

∏ Γ (ai + jAi )Γ (ai + (k − j)Ai ) Γ (ai + jAi )Γ (ai + (k − j)Ai ) , vkp,j = , Γ (bi − µ + jAi )Γ (bi + µ + (k − j)Ai ) Γ (bi + jAi )Γ (bi + (k − j)Ai ) i=1

and p

wkp,j =

uk,j

vkp,j

=

p ∏ i=1

Γ (bi + jAi )Γ (bi + (k − j)Ai ) Γ (bi − µ + jAi )Γ (bi + µ + (k − j)Ai )

166

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

Therefore,

wkp,j+1 p k,j

w

=

p ∏ Γ (bi + jAi + Ai )Γ (bi + (k − j)Ai − Ai )Γ (bi − µ + jAi )Γ (bi + µ + (k − j)Ai )

Γ (bi − µ + jAi + Ai )Γ (bi + µ + (k − j)Ai − Ai )Γ (bi + jAi )Γ (bi + (k − j)Ai ) [ ∏ Γ (bi + jAi + Ai )Γ (bi − µ + jAi ) ] [ Γ (bi + (k − j)Ai − Ai )Γ (bi + µ + (k − j)Ai ) ] = . Γ (bi + jAi )Γ (bi − µ + jAi + Ai ) Γ (bi + (k − j)Ai )Γ (bi + µ + (k − j)Ai − Ai ) i=1 p

(2.19)

i=1

By using the fact that the function z ↦ → Γ (z) is log-convex on (0, ∞), we deduce that the function z ↦ → and consequently, the following inequality

Γ (z +a) Γ (z)

is increasing

Γ (z + a + b)Γ (z) ≥ Γ (z + a)Γ (z + b),

(2.20)

is valid for all a, b > 0. In view of the above inequality, when z = bi − µ + jAi , a = µ and b = Ai , we obtain

Γ (bi + jAi + Ai )Γ (bi − µ + jAi ) ≥ 1. Γ (bi + jAi )Γ (bi − µ + jAi + Ai )

(2.21)

On the other hand, if we set z = bi + (k − j)Ai − Ai , a = Ai and b = µ, in (2.20), we find

Γ (bi + (k − j)Ai − Ai )Γ (bi + µ + (k − j)Ai ) ≥ 1. Γ (bi + (k − j)Ai )Γ (bi + µ + (k − j)Ai − Ai )

(2.22) p

So, combining (2.19), (2.21) and (2.22) we conclude that the sequence (wk,j )j≥0 is increasing, and consequently the sequence (Ak /Bk )k≥0 is increasing, by means of Lemma 1. Then, the function z ↦ → Φ (ap , bp , Ap , µ, z) is increasing on (0, ∞), from Lemma 2. This implies that

Φ (ap , bp , Ap , µ, z) ≥ Φ (ap , bp , Ap , µ, 0) =

p ∏ i=1

Γ 2 (bi ) . Γ (bi − µ)Γ (bi + µ)

This completes the proof of Theorem 3. □ Letting Ai = 1, in the above Theorem we get the following results: Corollary 1. Let µ > 0. If bi > max(µ, 1), then the function F (ap , bp , µ) defined by F (ap , bp , µ) = p Fp

[

a1 ,...,ap b1 −µ,...,bp −µ

⏐ ])2 ⏐ ]/( [ ⏐ ] [ a1 ,...,ap ⏐ a1 ,...,ap ⏐ ⏐ , z F ⏐ ⏐z p Fp b1 +µ,..., p p b1 ,...,bq ⏐z bp +µ

is increasing on (0, ∞). Moreover, the following Turán-type inequality p Fp

[

a1 ,...,ap b1 −µ,...,bp −µ

⏐ ])2 ⏐ ] ( [ ⏐ ] [ a1 ,...,ap ⏐ a1 ,...,ap ⏐ ⏐ ≥ 0, ⏐z p Fp b1 +µ,..., bp +µ ⏐z − p Fp b1 ,...,bq ⏐z

(2.23)

holds true for all z > 0. 3. Sub-additivity and super-additivity properties for the Fox-Wright functions In the following Lemma we present a necessary condition for a function to be sub-additive. For more details, one may see [27, Theorem 15.4, p. 226]. Let f : (0, ∞) −→ R be a function. If

Lemma 3. Theorem 4.

f (x) x

is decreasing, then f is sub-additive.

The function

b 1 ↦ → p Ψp

[

(ap +Ap ,Ap ) (bp ,Ap )

⏐ ]/ [ ⏐ ] (ap ,Ap ) ⏐ ⏐ ⏐z p Ψp (bp ,Ap ) ⏐z ,

is decreasing and sub-additive on (0, ∞). Moreover, the following inequality (a1 +A1 ,A1 ),...,(ap +Ap ,Ap ) p Ψp (b +b′ ,A ),(b ,A ),...,(b ,A ) p p 1 2 2 1 1

⏐ ] ⏐ ] [ (a1 +A1 ,A1 ),...,(ap +Ap ,Ap ) ⏐ ⏐ ⏐z ⏐z p Ψp (b1 ,A1 ),...,(bp ,Ap ) ⏐ ] ⏐ ] [ [ ≤ (a1 ,A1 ),...,(ap ,Ap ) (a1 ,A1 ),...,(ap ,Ap ) ⏐ ⏐ p Ψp (b +b′ ,A ),...,(b ,A ) ⏐z p Ψp (b1 ,A1 ),...,(bp ,Ap ) ⏐z p p 1 1 1 ⏐ ] [ (a1 +A1 ,A1 ),...,(ap +Ap ,Ap ) ⏐ p Ψp (b′ ,A ),(b ,A ),...,(b ,A ) ⏐z p p ⏐ ] [ 1 1 2 2 + (a1 ,A1 ),...,(ap ,Ap ) ⏐ Ψ p p (b′ ,A ),(b ,A ),...,(b ,A ) ⏐z p p 1 2 2 [

1

is valid for all z , bi , b′1 , ai , Ai > 0.

(3.1)

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

167

Proof. We assume that z > 0. We consider the following auxiliary function f (b1 ) = p Ψp

(ap +Ap ,Ap ) (bp ,Ap )

[

⏐ ]/ [ ⏐ ] (ap ,Ap ) ⏐ ⏐ ⏐z p Ψp (bp ,Ap ) ⏐z .

The Cauchy product rule gives (ap ,Ap ) p Ψp (bp ,Ap )

(

[

⏐ ])2 ∂ f (b ) ⏐ ] ∂ ⏐ ] ⏐ ] ⏐ ] ∂ [ [ [ [ (a ,A ) ⏐ (ap +Ap ,Ap ) ⏐ (ap ,Ap ) ⏐ (ap +Ap ,Ap ) ⏐ ⏐ 1 = p Ψp (bpp ,App ) ⏐z ⏐z p Ψp p Ψp (bp ,Ap ) ⏐z (bp ,Ap ) ⏐z − p Ψp (bp ,Ap ) ⏐z ∂ b1 ∂ b1 ∂ b1 )( ∞ p ) (∞ p ∑ ∏ Γ (ai + kAi + Ai )ψ (b1 + kA1 ) z k ∑ ∏ Γ (ai + kAi ) z k − = Γ (bi + kAi ) k! Γ (bi + kAi ) k! =0 i=1 k=0 i=1 (∞ p )k( ) p ∞ ∏ ∑ ∏ Γ (ai + kAi + Ai ) z k ∑ Γ (ai + kAi )ψ (b1 + kA1 ) z k + Γ (bi + kAi ) k! Γ (bi + kAi ) k! =

k=0 i=1 p ∞ k ∑∑ ∏ k=0 j=0 i=1

−

k=0 i=1

zk Γ (ai + jAi + Ai )Γ (ai + (k − j)Ai )ψ (b1 + (k − j)A1 ) Γ (bi + jAi )Γ (bi + (k − j)Ai ) j!(k − j)!

p ∞ ∑ k ∑ ∏ Γ (ai + (k − j)Ai + Ai )Γ (ai + jAi )ψ (b1 + (k − j)A1 )

Γ (bi + (k − j)Ai )Γ (bi + jAi )

k=0 j=0 i=1

=

(3.2)

zk j!(k − j)!

(i)

p ∞ ∑ k ∑ ∏

Tk,j z k

k=0 j=0 i=1

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

,

where (i)

Tk,j = ψ (b1 + (k − j)A1 )[Γ (ai + jAi + Ai )Γ (ai + (k − j)Ai ) − Γ (ai + (k − j)Ai + Ai )Γ (ai + jAi )]. If k is even, then k

(i)

p k ∑ ∏

Tk,j z k

j=0 i=1

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

=

−1 p 2 ∑ ∏

Tk,j z k

j=0 i=1

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

(i)

(i)

p

k

Tk,j z k

∑ ∏

+

j= 2k +1 i=1

[

k−1

(3.3)

]

2 p ∑ ∏

=

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

j=0

i=1

(i)

(i)

(Tk,j + Tk,k−j )z k

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

where [.] denotes the greatest integer function. Similarly, if k is odd, then [

∑∏

(i) Tk,j z k

j=0 i=1

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

k

p

=

k−1 2

]

p ∑ ∏ j=0

i=1

(i)

(i)

(Tk,j + Tk,k−j )z k

Γ (bi + (k − j)Ai )Γ (bi + jAi )j!(k − j)!

(3.4)

Simplifying, we get (i)

(i)

Tk,j + Tk,k−j = (ψ (b1 + (k − j)A1 ) − ψ (b1 + jA1 ))

× (Γ (ai + jAi + Ai )Γ (ai + (k − j)Ai ) − Γ (ai + (k − j)Ai + Ai )Γ (ai + jAi ))

(3.5)

For k − j > j (i.e. for [ k−2 1 ] ≥ j), and using the fact that the digamma function ψ is increasing on (0, ∞), we get

(ψ (b1 + (k − j)A1 ) − ψ (b1 + jA1 )) ≥ 0.

(3.6)

On the other hand, letting z = ai + jAi , a = Ai and b = (k − 2j)Ai in (2.20), we have

Γ (ai + (k − j)Ai + Ai )Γ (ai + jAi ) ≥ Γ (ai + jAi + Ai )Γ (ai + (k − j)Ai ).

(3.7)

In view of (3.2), (3.5), (3.6) and (3.7) we deduce that the function b1 ↦ → f (b1 ) is decreasing and consequently the function f (b ) b1 ↦ → b 1 , is decreasing, as a product of two positive and decreasing functions. So, Lemma 3 completes the proof of this 1 Theorem. □ Letting Ai = 1 in Theorem 4, we obtain the following result for the hypergeometric function p Fp [.].

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K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

Corollary 2. The function p Fp

a1 +1,...,ap +1 b1 ,...,bp

[

b1 ↦ →

[

p Fp

a1 ,...,ap b1 ,...,bp

]

]

,

is decreasing and sub-additive on (0, ∞). Furthermore, the following inequality p Fp

[

p Fp

a1 +1,...,ap +1

]

p Fp

b1 +b′1 ,b2 ...,bp

[

] ≤

a1 ,...,ap

[

a1 +1,...,ap +1 b1 ,...,bp

p Fp

b1 +b′1 ,...,bp

[

a1 ,...,ap b1 ,...,bp

]

p Fp

[

+

]

a1 +1,...,ap +1 b1 ,...,bp

p Fp

a1 ,...,ap

[

]

]

,

(3.8)

b′1 ,...,bp

is valid. Theorem 5. Let a, b, A and B be a real numbers such that a, A, B > 0 and b ≥ 1. The following assertions are true: 1. If b−2 1 < a < 2b , the function a ↦→

1

Γ (a)Γ (b − a)

.2 Ψ1

[

(a,A),(b−a,A) (b,B)

⏐ ] ⏐ ⏐z , z > 0,

is sub-additive. In addition the following inequality 1

Γ (a1 + a2 )Γ (b − a1 − a2 )

.2 Ψ1

(a1 +a2 ,A),(b−a1 −a2 ,A) (b,B)

[

1

⏐ ] ⏐ ⏐z ≤

⏐ ] ⏐ ⏐z Γ (a1 )Γ (b − a1 ) ⏐ ] [ 1 (a ,A),(b−a ,A) ⏐ .2 Ψ1 2 (b,B) 2 ⏐z , + Γ (a2 )Γ (b − a2 )

holds true for all a1 , a2 ∈ (b − 1/2, b/2) and z > 0.

.2 Ψ1

p,0

2. Let µ > 0, γ = min1≤i≤p (ai /Ai ) ≥ 1 and ∆ = 0. If the Fox H-function Hq,p

∏q

j=1

Γ (bj )

i=1

Γ (ai )

z ↦ → log ∏p

p Ψq

[

(ap ,Ap ) (bq ,Bq )

(Bp ,bp ) (Ap ,ap )

[

[

(a1 ,A),(b−a1 ,A) (b,B)

(3.9)

⏐ ] ⏐ ⏐z is non-negative, then the function

⏐ ] ⏐ ⏐−z ,

is super-additive on (0, ∞). Furthermore, the following inequality

∏q

j=1

∏p

i=1

Γ (bj ) Γ (ai )

p Ψq

[

(ap ,Ap ) (bp ,Bp )

⏐ ⏐ ⏐ ] ] [ ] [ (ap ,Ap ) ⏐ (ap ,Ap ) ⏐ ⏐ ⏐ − z1 p Ψq (bp ,Bp ) ⏐ − z2 ≤ p Ψq (bp ,bp ) ⏐ − z1 − z2 ,

(3.10)

is valid for all z1 , z2 > 0. Proof. 1. For convenience, let us write

Ξa [b, A, B|z] :=

1

Γ (a)Γ (b − a)

.2 Ψ1

[

(a,A),(b−a,A) (b,B)

∞ ⏐ ] ∑ ⏐ ⏐z = k=0

fk (a) k!Γ (b + kB)

zk,

where, fk (a) =

Γ (a + kA)Γ (b − a + kA) . Γ (a)Γ (b − a)

It is enough to show that the function a ↦ → gk (a) =

fk (a) a

is decreasing. Indeed, straightforward calculation would yield

∂ gk (a) = gk (a) [(ψ (a + kA) − ψ (b − a + kA)) + (ψ (b − a) − ψ (a + 1))] . ∂a Using the fact that the digamma function ψ is increasing on (0, ∞), we obtain

(3.11)

ψ (a + kA) − ψ (b − a + kA) ≤ 0, b ≥ 2a, ψ (b − a) − ψ (a + 1) ≤ 0, b ≤ a + 1. This implies that the function a ↦ → gk (a) is decreasing and consequently a ↦ → fk (a) is sub-additive. Then, for all b − 1/2 < a1 , a2 < b/2 and z > 0 we obtain

Ξa1 +a2 [b, A, B|z] =

≤

∞ ∑ fk (a1 + a2 )z k k=0

k!Γ (b + kB)

∞ ∑

fk (a1 )z k

k=0

k!Γ (b + kB)

+

∞ ∑

fk (a2 )z k

k=0

k!Γ (b + kB)

= Ξa1 [b, A, B|z] + Ξa2 [b, A, B|z] .

(3.12)

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

169

2. By using [21, Theorem 1], the function

∏q

j=1

Γ (bj )

i=1

Γ (ai )

z ↦ → ∏p

p Ψq

(ap ,Ap ) (bp ,bp )

[

⏐ ] ⏐ ⏐−z ,

is completely monotonic on (0, ∞) and maps (0, ∞) into (0, 1). According to a result of Kimberling [28], if a function f , defined on (0, ∞), is continuous and completely monotonic and maps (0, ∞) into (0, 1), then log f is super-additive, that is for all x, y > 0 we have f (x)f (y) ≤ f (x + y). Which completes the proof of the second assertion.

□

Corollary 3. We consider the τ −Gauss hypergeometric function 2 ϕ1τ (a, b, c , z) defined by [29]: ∞

τ 2 ϕ1 (a, b, c , z) =

Γ (c) ∑ Γ (a + k)Γ (b + τ k) z k , a, b, c > 0, c > b, |z | < 1. Γ (a)Γ (b) Γ (c + τ k) k! k=0

τ

Then, the function a ↦ → 2 ϕ1 (a, c − a, c , z) is sub-additive, and satisfies the following inequality

ϕτ

2 1 (a1

+ a2 , c − a1 − a2 , c , z) ≤ 2 ϕ1τ (a1 , c − a1 , c , z)2 ϕ1τ (a2 , c − a2 , c , z).

(3.13)

⏐ a,b−a ⏐ is valid for all a1 , a2 ∈ ((c − 1)/2, c /2). In particular, the Gauss hypergeometric function a ↦ → 2 F1 [ b ⏐z ] is sub-additive. Proof. Follows by Theorem 5, when choosing A = B = τ and A = B = 1, respectively.

□

Here we only demonstrate the direct consequences of Theorem 5 when it is combined with the results of Mehrez [30]. Corollary 4. Under the conditions (H1 ) : α ∈ (0, 1), d − α ≥ 1, β ≥

d 2

1

+ , d ∈ {2, 3, . . .} . 2

The function

{ z ↦ → log

Γ (β − α ) (d α) ( ) 2 Ψ1 Γ 2 − 2 Γ 1 − α2

[

( 2d − α , 1 ),(1− α2 , 12 ) 2 2 (β−α,1)

]} ⏐ ⏐ ⏐−z ,

is super-additive. Moreover the following inequality

Γ

(d 2

Γ (β − α ) ) ( ) 2 Ψ1 − α2 Γ 1 − α2

[

( 2d − α , 1 ),(1− α2 , 12 ) 2 2 (β−α,1)

[d α 1 ] ] ⏐ ⏐ ( − , 2 ),(1− α ,1) ⏐ 2 2 ⏐ − z ⏐ − z1 .2 Ψ1 2 2(β−α, ⏐ 2 1) [d α 1 ] ⏐ ( − , 2 ),(1− α ,1) 2 2 ⏐ ≤ 2 Ψ1 2 2(β−α, − z − z ⏐ 1 2 , 1)

holds true for all z1 , z2 > 0. Proof. The claim follows from Theorem 5 combining with [30, Corollary 1].

□

A combination of [30, Corollary 3] with the second assertions in Theorem 5 immediately yields: Corollary 5. Assume that the hypotheses

⎧ √ ⎪ [3 − 5 ) ⎪ ⎪ γ ∈ R , α ∈ ,1 , ⎪ ⎪ ⎪ 2 ⎨ ( ( )) (H2 ) : 1 min d − 2, 2 γ + 1−α ≥ 1, ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ 5 > 1 + γ (1 − α ) + d , d ∈ {3, 4, . . .} . 2 α 2 hold true. Then the function z ↦ → log

⎧ ⎨

Γ (1 + γ (1 − α )) ( ) 2 Ψ1 ⎩ Γ ( d − 1) Γ 1+γ (1−α) 2 α

[

1+γ (1−α ) 1−α ( 2d −1, 21 ),( , 2α ) α (1+γ (1−α ), 21α )

⎫ ]⎬ ⏐ ⏐ , ⏐−z ⎭

(3.14)

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K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

is super-additive. For every z1 , z2 > 0, it holds

Γ (1 + γ (1 − α )) ) 2 Ψ1 (d ) ( Γ 2 − 1 Γ 1+γ α(1−α)

[

1+γ (1−α ) 1−α , 2α ) α (1+γ (1−α ), 21α )

( 2d −1, 12 ),(

] [ d 1 1+γ (1−α) 1−α ⏐ ] ⏐ ( 2 −1, 2 ),( , 2α ) ⏐ ⏐ α ⏐ − z1 .2 Ψ1 ⏐ − z2 (1+γ (1−α ), 1 ) 2α

(3.15)

≤ 2 Ψ1

[

1+γ (1−α ) 1−α , 2α ) α (1+γ (1−α ), 21α )

( 2d −1, 21 ),(

] ⏐ ⏐ ⏐ − z1 − z2 .

Corollary 6. Under the conditions

⎧ ⎪ ⎪ ⎪ α ∈ (0, 1], γ ∈ R, 2(γ + β ) ≥ 1 ⎪ ⎪ ⎪ ⎨ 1 1 1 =1+ , − 1 < β, (H3 ) : α αβ α ⎪ ⎪ ( ) ⎪ ⎪ ⎪ γ 1 1 1 ⎪ ⎩1 + 1− =1+ > , β α αβ α the function

) ⎧ ( ⎫ [ ]⎬ ⎨ Γ 1 + γβ ⏐ γ +β 1 (1,1),( αβ , 2αβ ) ⏐ ( ) 2 Ψ2 z ↦ → log , ⏐−z γ (1+ β , 21α ),(1, 12 ) ⎩ Γ γ +β ⎭ αβ is super-additive. For z1 , z2 > 0, it holds:

( ) [ ] ] [ ⏐ ⏐ γ +β γ +β Γ 1 + γβ 1 1 (1,1),( αβ , 2αβ (1,1),( αβ , 2αβ ) ⏐ ) ⏐ ( ) 2 Ψ2 ⏐ − z1 .2 Ψ2 ⏐ − z2 γ γ (1+ β , 21α ),(1, 12 ) (1+ β , 21α ),(1, 12 ) +β Γ γαβ [ ] ⏐ γ +β 1 (1,1),( αβ , 2αβ ) ⏐ ≤ 2 Ψ2 − z − z . ⏐ 1 2 γ 1 1

(3.16)

(1+ β , 2α ),(1, 2 )

Proof. The claim follows immediately by combining the second result in Theorem 5 with [30, Corollary 4]. □ 4. Some geometric properties for the Fox-Wright functions The main purpose of this section is to investigate certain criteria for the univalence and starlikeness for the Fox–Wright function p+1 Ψp [.]. Each of the following definitions will be used in the remainder of our investigation. Definition 1. A function f is said to be univalent in a domain D if it is one-to-one in D. An analytic function f is said to be starlike (with respect to the origin 0), if t w ∈ f (D) whenever w ∈ f (D) and t ∈ [0, 1]. Theorem 6. function

Let αi , βi be a real numbers, such that the hypotheses (H) are satisfied. Suppose that 0 < σ ≤ 2. Then, the

z ↦ → z p+1 Ψp

[

(σ ,1),(αp ,A) (βp ,A)

⏐ ] ⏐ ⏐z

is univalent in the disk |z | < rs :=

√√

32 − 5 ≈ 0.81.

Proof. Combining [21, Theorem 4] with [31, Satz 3.2] we obtain the desired result. □ Theorem 7. Let 0 < σ ≤ 1. Assume that the hypotheses (H) of Theorem 6 are satisfied. Then the following assertions are true: 1. The functions z ↦→

p+1 Ψp

(σ ,1),(αp ,A) (βp ,A)

[

⏐ ] ⏐ ] [ (σ ,1),(αp ,A) ⏐ ⏐ ⏐z , and z ↦→ z p+1 Ψp (βp ,A) ⏐z ,

are univalent in the half-plane ℜ(z) < 1. 2. The second function is starlike in the disk |z | < r ∗ , where

√

r ∗ :=

5− 12

13

√

√

10 13 − 2 +

√

13 − 3 3

√

√

13 − 2 ≈ 0.934.

K. Mehrez / Journal of Computational and Applied Mathematics 362 (2019) 161–171

171

Proof. The proof of the first claim follows from Stieltjes transform representation for the Fox–Wright functions [21, Theorem 4] with Theorem 2.1 and Theorem 2.2 in [32]. A combination of Theorem 4 in [21] with [31, Satz 2.4] immediately yields the second claim. □ Setting A = τ in Theorems 6 and 7 immediately yields the following geometric properties for the τ −Gauss hypergeometric function 2 ϕ τ (a, b, c ; z). Corollary 7. Let a, b, c be a positive real numbers such that c > b. The following properties are true: 1. The functions z ↦ → 2 ϕ1τ (a, b, c , z); and z ↦ → z .2 ϕ1τ (a, b, c , z), are univalent in the half-plane ℜ(z) < 1. 2. The second function is starlike in the disk |z | < r ∗ and also univalent in the disk |z | < rs . Remark 2. Theorems 6 and 7 improve on Theorem 13 and Theorem 14 in [33]. 5. Conclusion Lastly, we conclude this paper by remarking that, we have obtained several new properties for some classes of functions related to the Fox–Wright functions. In particular, the monotonicity, convexity, sub-additivity and super-additivity, as well as Turán-type inequalities, are established in more general cases when p = q. In addition, a brief consideration of the criteria for univalence and starlikeness of the Fox–Wright function when q = p + 1 are given. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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