UNIVALENT FUNCTIONS RELATED TO q–ANALOGUE OF GENERALIZED M –SERIES WITH RESPECT TO k–SYMMETRIC POINTS

ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 14 (2019), 277 – 285

UNIVALENT FUNCTIONS RELATED TO q–ANALOGUE OF GENERALIZED M –SERIES WITH RESPECT TO k–SYMMETRIC POINTS

Sh. Najafzadeh

Abstract. In this paper, we introduce subclasses of analytic functions by using q–analogue of generalized M –series and k–symmetric points. Some special coefficient inequalities are also discussed.

1 Introduction

The M –series in [8] in given by:

α

tM_s(z) =

∞

∑

k=0

(d1)k· · · (d_t)k

(b1)_k· · · (b_s)_k z^k

Γ(αk + 1), (1.1)

where α, z ∈ C, Re{α} > 0 and (dm)k, (bm)k are then well-known Pochhammer symbols.

It is easy to see that by the ratio test the series in (1.1) is convergent for all z if t 6 s.

The extension of both Mittag-Leffler function and generalized hypergeometric function_xF_y is called generalized M –series which introduced in [9] as follow:

α tM^β_s =

∞

∑

k=0

(d₁)_k· · · (d_t)_k (b₁)_k· · · (b_s)_k

z^k

Γ(αk + β), (z, α, β ∈ C). (1.2) For more details see [1].

The series in [2] is a convergent series for all z if t 6 s + Re{α}. Also it is convergence for |z| < α^α if t = s + Re{α}.

The q–analogue of Pochhammer symbol is defined by:

(α; q)n=

n−1

∏

k=0

(1 − αq^k), (n ∈ N), (1.3)

2010 Mathematics Subject Classification: 30C45; 30C50.

Keywords: M –series; q–derivative; univalent function; convolution; k–symmetric points.

and for n = 0 and q ̸= 1, (α; q)₀ = 1. When n → ∞, we shall assume that |q| < 1, see [2].

Also q–derivative of a function f (z) is defined by (D_qf)(z) = f (z) − f (qz)

(1 − q)z , (z ̸= 0, q ̸= 1), (1.4) and

q→1limDqf (z) = f^′(z). (1.5) By using (1.4), we conclude that:

(Dⁿ1 q

f)(x) = qⁿ(Dⁿ_qf)( x

qⁿ), (1.6)

Dⁿ_qz^λ = Γ_q(λ + 1)

Γq(λ − n + 1)z^λ−n, ( Re{λ} + 1 > 0). (1.7) Indeed

Γ_q(z + 1) = 1 − q^z

1 − z Γ_q(z). (1.8)

Also

β_q(x, y) =

∫ 1 0

t^x−1 (tq; q)∞

(tq^y; q)∞

dq(t) = Γ_q(x)Γ_q(y)

Γq(x + y) , (1.9)

where Re{x} > 0, Re{y} > 0 and βq(x, y) and Γq(w) are the q–analogue of the beta function and q–gamma function respectively.

Now we consider the q–analogue of generalized M –series as follows:

α

tM^β_s(z; q) =

∞

∑

k=0

(d1; q)k· (d_t; q)k

(b1; q)_k· · · (b_s; q)_k(q; q)_k

z^k

Γq(αk + β), (1.10) where α, β ∈ C, Re{α} > 0, |q| < 1, (γ; q)kis the q–analogue of Pochhammer symbol and Γ_q(w) is the q–gamma function.

By applying the convergent conditions of the well-known Fox-Wright generalized hypergeometric function and generalized H–function, the function ^α_tM^β_s(z; q) is convergent, see [3].

Some special eases of ^α_tM^β_s(z; q) are:

(i) The q–Mittag-Leffler function [5].

(ii) The generalized q–Mittag-Leffler function [10].

(iii) The q–generalized M –series as a special ease of the q–Wright generalized hypergeometric function [6].

Definition 1. Let A denote the class of functions f (z) of the type:

f (z) = z +

∞

∑

k=2

a_kz^k, (1.11)

which are analytic in the open unit disk:

U = {z ∈ C : |z| < 1}.

Definition 2. the Hadamard product (convolution) for functions f (z) = z+∑∞ k=2a_kz^k and g(z) = z +∑∞

k=2b_kz^k belong to A denoted by f ∗ g in defined as follows:

(f ∗ g)(z) = z +

∞

∑

k=2

akbk = (g ∗ f )(z). (1.12)

Definition 3. A function f (z) ∈ A is in the class Xn(θ) if

Re

{z(f ∗ F )^′+ (f ∗ F ) H_n(z)

}

< θ, (1.13)

where θ > 1, n > 1 is a fixed positive integer,

F (z) =[

1 − (1 − d1) · · · (1 − dt) (1 − b₁) · · · (1 − b₁)Γ(α + β)

]

z − 1

Γ_q(β)+^α_tM^β_s(z; q), (1.14) and

Hn(z) = 1 n

n−1

∑

v=0

E^−v(f ∗ F )(E^vz), (Eⁿ= 1, z ∈ U). (1.15)

Further, a function f (z) ∈ A is in the class Yn(θ), if and only if zf^′(z) ∈ Xn(θ).

From (1.11), (1.14) and (1.12) with a simple calculation, we get:

(f ∗ F )(z) = z +

∞

∑

k=2

(d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (b_s; q)_k(q; q)_kΓq(αk + β)akz^k. (1.16)

2 Main results

In this section, we shall obtain some coefficient bounds for functions in Xn(θ) and Y_n(θ) and their subclasses positive coefficients.

Note that other subclasses of analytic functions with respect to n–symmetric points have been studied by many authors, see [4,7] and [11].

Theorem 4. Let θ > 2. If f (z) ∈ A satisfies:

∞

∑

k=1

[ (nk + 2)(d1; q)_nk+1· · · (d_t; q)_nk+1

(b₁; q)_nk+1· · · (b_s; q)_nk+1(q; q)_nk+1Γ(α(nk + 1) + β) +

⏐

⏐

⏐

⏐

⏐

(nk + 2)(d₁; q)_nk+1· · · (d_t; q)_nk+1

(b₁; q)_nk+1· · · (b_s; q)_nk+1(q; q)_nk+1Γ(α(nk + 1) + β) − 2θ

⏐

⏐

⏐

⏐

⏐ ]

|a_nk+1|

+

∞

∑

n=2 k̸=mk+1

2(k + 1)(d1; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ(αk + β)|a_k| 6 2(θ − 2),

(2.1)

then f ∈ Xn(θ).

Proof. Suppose that theta > 2 and f (z) ∈ A, it is sufficient to show that:

⏐

⏐

⏐

⏐

z(f ∗ F )^′+ (f ∗ F ) f_n(z)

⏐

⏐

⏐

⏐

<

⏐

⏐

⏐

⏐

z(f ∗ F )^′+ (f ∗ F ) f_n(z) − 2θ

⏐

⏐

⏐

⏐

, (z ∈ U).

By putting W =⏐

⏐z(f ∗ F )^′+ (f ∗ F )⏐

⏐−⏐

⏐z(f ∗ F )^′+ (f ∗ F ) − 2θf_n(z)⏐

⏐, and

Hn(z) = z + 1 n

∞

∑

k=2

(d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (b_s; q)_k(q; q)_kΓ(αk + β)αk n−1

∑

v=0

E^v(k−1)zⁿ, (2.2) we get

W =

⏐

⏐

⏐

⏐

⏐ 2z +

∞

∑

k=2

(k + 1) (d1; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ_q(αk + β)a_kz^k

⏐

⏐

⏐

⏐

⏐

−

⏐

⏐

⏐

⏐

⏐ 2z +

∞

∑

k=2

(k + 1) (d₁; q)_k· · · (d_t; q)_ka_kz^k (b₁; q)_k· · · (b_s; q)_k(q; q)_k(αk + β; q)_k

− 2θz − 2θ

∞

∑

k=2

(d₁; q)_k· · · (d_t; q)_ka_kb_kz^k (d1; q)k· · · (b_s; q)k(q; q)kΓq(αk + β)

⏐

⏐

⏐

⏐

⏐ ,

where

b_k= 1 n

n−1

∑

v=0

E^v(k−1), (Eⁿ= 1). (2.3)

Hence for |z| = r < 1, we have:

W 6 2r +

∞

∑

k=2

(k + 1)(d₁; q)_k· · · (d_t; q)_ka_kr^k (b1; q)k· · · (b_s; q)k(q; q)kΓ(αk + β)

− {

2(θ − 1)r −

∞

∑

k=2

⏐

⏐

⏐

⏐

(k + 1)(d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (b_s; q)_k(q; q)_kΓ(αk + β)− 2θb_k

⏐

⏐

⏐

⏐

|a_k|r^k }

<

{ _∞

∑

k=2

[ (k + 1)(d1; q)_k· · · (d_t; q)_k (b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ(αk + β) +

⏐

⏐

⏐

⏐

⏐

(k + 1)(d₁; q)_k· · · (d_t; q)_k

(b1; q)k· · · (b_s; q)k(q; q)kΓ(αk + β)− 2θb_k

⏐

⏐

⏐

⏐

⏐ ]

|a_k| − 2(θ − 2) }

r.

From (2.3) we know

bk =

{0 , k ̸= mn + 1

1 , k = mn + 1. (2.4)

So we get

W <

{ _∞

∑

k=1

[ (nk + 2)(d1; q)_nk+1· · · (d_t; q)_nk+1

(b₁; q)_nk+1· · · (b_s; q)_nk+1(q; q)_nk+1Γ(α(nk + 1) + β) +

⏐

⏐

⏐

⏐

⏐

(nk + 2)(d₁; q)_nk+1· · · (d_t; q)_nk+1

(b₁; q)_nk+1· · · (b_s; q)_nk+1(q; q)_nk+1Γ(α(nk + 1) + β) − 2θ

⏐

⏐

⏐

⏐

⏐ ]

|a_nk+1|

+

∞

∑

k=2 k̸=mn+1

2(k + 1)(d₁; q)_k· · · (d_t; q)_k

(b1; q)k· · · (b_s; q)k(q; q)kΓ(αk + β)|a_k| − 2(θ − 2) }

r.

From (2.1) we know that W < 0. Thus we get then required result.

By definition of Y_n(θ) we obtain the following corollary.

Corollary 5. If θ > 2 and (f ∗ F )(z) is defined by (1.16) satisfies

∞

∑

k=0

(nk + 2)

[ (nk + 2)(d₁; q)_nk+1· · · (d_t; q)_nk+1

(b₁; q)_nk+1· · · (b_s; q)_nk+1(q; q)_nk+1Γ(α(nk + 1) + β) +

⏐

⏐

⏐

⏐

⏐

(nk + 2)(d₁; q)_nk+1· · · (d_t; q)_nk+1

(b1; q)nk+1· · · (b_s; q)nk+1(q; q)nk+1Γ(αk + 1) + β) − 2θ

⏐

⏐

⏐

⏐

⏐ ]

|a_nk+1|

+

∞

∑

k̸=mk+1k=2

2(k + 1)²(d₁; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ(αk + β)|a_k| 6 2(θ − 2),

(2.5)

then f (z) ∈ Y_n(θ).

Now, we define two subclasses of Xn(θ) and Yn(θ) as follow:

X_n⁺(θ) ={

f ∈ X_n(θ) : The coefficients of f ∗ F are non-negative}

, (2.6) Y_n⁺(θ) ={

f ∈ Y_n(θ) : The coefficients of f ∗ F are non-negative}

, (2.7)

Theorem 6. Let n > 2 and 2 < θ 6 n + 1, then f (z) ∈ Xn⁺(θ) if and only if

∞

∑

k=2

(k + 1)(d1; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ_q(αk + β)a_k

− θ

∞

∑

m=1

(d1; q)mn+1· · · (d_t; q)mn+1amn+1

(b1; q)mn+1· · · (b_s; q)mn+1(q; q)mn+1Γq(α(mn + 1) + β) 6 2(θ − 2).

(2.8)

Proof. According to Theorem4, we need only to prove the necessary. Since f (z) ∈ X_n⁺(θ), then

(d₁; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ_q(αk + β)a_k> 0, (k > 2).

Also

Re{z(f ∗ F )^′+ (f ∗ F ) Hn(z)

}

< θ, or equivalently

⏐

⏐

⏐

⏐

z(f ∗ F )^′+ (f ∗ F ) Hn(z)

⏐

⏐

⏐

⏐

<

⏐

⏐

⏐

⏐

z(f ∗ F )^′+ (f ∗ F ) Hn(z)

⏐

⏐

⏐

⏐ , or

⏐⏐z(f ∗ F )^′+ (f ∗ F )⏐

⏐<⏐

⏐z(f ∗ F )^′+ (f ∗ F ) − 2θH_n(z)⏐

⏐.

Hence

⏐

⏐

⏐

⏐

⏐ z +

∞

∑

k=2

k(d1; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (b_s; q)_k(q; q)_kΓ_q(αk + β)a_kz^k + z +

∞

∑

k=2

(d₁; q)_k· · · (d_t; q)_k

(b1; q)k· · · (q; q)_kΓq(αk + β)a_kz^k

⏐

⏐

⏐

⏐

⏐

<

⏐

⏐

⏐

⏐

⏐ z +

∞

∑

k=2

k(d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (q; q)_kΓq(αk + β)akz^k + z +

∞

∑

k=2

(d₁; q)_k· · · (d_t; q)_k

(b1; q)k· · · (q; q)_kΓq(αk + β)a_kz^k− 2θ

− 2θ

∞

∑

m=1

(d1; q)mn+1· · · (d_t; q)mn+1amn+1

(b1; q)mn+1· · · (q; q)_mn+1Γq(α(mn + 1) + β)z^mn+1

⏐

⏐

⏐

⏐

⏐ . Since

(d1; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (q; q)_kΓ_q(αk + β)a_k> 0, for k > 2 and θ > 2, by setting z → 1⁻, we get:

2 +

∞

∑

k=2

(k + 1) (d₁; q)_k· · · (d_t; q)_k

(b₁; q)_k· · · (q; q)_kΓ_q(αk + β)a_k6 2θ − 2 + 2θ

∞

∑

m=1

(d₁; q)_mn+1· · · (d_t; q)_mn+1

(b₁; q)_mn+1· · · (q; q)_mn+1Γ_q(α(mn + 1) + β)a_mn+1

−

∞

∑

k=2

(k + 1) (d₁; q)_k· · · (d_t; q)_k

(b1; q)k· · · (q; q)_kΓq(αk + β)a_k, or

∞

∑

k=2

(k + 1) (d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (q; q)_kΓq(αk + β)ak

− θ

∞

∑

m=1

(d1; q)mn+1· · · (d_t; q)mn+1amn+1

(b1; q)mn+1· · · (q; q)_mn+1Γq(α(mn + 1) + β) 6 2(θ − 2).

Similarly, we have the following theorem for the class Y_n(θ).

Corollary 7. Let n > 2, 2 < θ 6 n + 1 and f (z) ∈ A, then f (z) is in the class Y_n(θ) if and only if

∞

∑

k=2

(k + 1)² (d1; q)k· · · (d_t; q)k

(b1; q)_k· · · (q; q)_kΓq(αk + β)ak

− θ

∞

∑

m=1

(mn + 1) (d1; q)mn+1· · · (d_t; q)mn+1amn+1

(b₁; q)_mn+1· · · (q; q)_mn+1Γ_q(α(mn + 1) + β) 6 2(θ − 2).

References

[1] A. Chouhan and S. Saraswat, Certain properties of fractional calculus operators associated with M –series, Sci., Ser. A, Math. Sci., 22 (2012), 27–32.MR3058917.

Zbl 1258.33013.

[2] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge university press 96, 2004. MR2128719.Zbl 1129.33005.

[3] S. H. Malik, R. Jain and J. Majid, Certain properties of q–fractional integral operators associated with q–analogue of generalized M –series, UGC Approved Journal, 12(4) (2019), 1006–1015.

[4] S. Najafzadeh, Application of S˘al˘agean operator on univalent functions with respect to k–symmetric points, Adv. Math., Sci. J., 1 (2016), 39–43. Zbl 1377.30013.

[5] S. D. Purohit and F. Ucar, An application of q–sumudu transform for fractional q–kinetic equation, Turk. J. Math., 42(2) (2018), 726–734. MR3794501. Zbl 07051495.

[6] R. Saxena and R. Kumar, A basic analogue of the generalized H–function, Matematiche (Catania), 50(2) (1995), 263–271. MR1414634.Zbl 0899.33012.

[7] C. Selvaraj, K. Karthikeyan and E. Umadevi, Certain classes of meromorphic multivalent functions with respect to (j, k)–symmetric points, An. Univ. Oradea, Fasc. Mat., 20(2) (2011), 173–180. Zbl 1313.30078.

[8] M. Sharma, Fractional integration and fractional differentiation of the M –series, Fract. Calc. Appl. Anal., 11(2) (2008), 187–191.MR2401326.Zbl 1153.26303.

[9] M. Sharma and R.Jain, A note on a generalized M –series as a special function of fractional calculus, Fract. Calc. Appl. Anal., 12(4) (2009), 449–452.MR2598192.

Zbl 1196.26013.

[10] S. Sharma and R. Jain, On some properties of generalized q–Mittag-Leffler function, Mathematica Aeterna, 4(6) (2014), 613–619.

[11] Z. Wang, C. Gao and S. Yuan, Some coefficient inequalities for certain subclasses of analytic functions with respect to k–symmetric points, Soochow J.

Math., 33(2) (2007), 223–227. MR2330472.Zbl 1130.30015.

Sh. Najafzadeh

Department of Mathematics, Payame Noor University, Post Office Box: 19395–3697, Tehran, Iran.

e-mail: najafzadeh1234@yahoo.ie.

License

This work is licensed under aCreative Commons Attribution 4.0 International License.