operator to analytic functions of reciprocal order

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DOI: 10.24193/subbmath.2020.2.03

Application of Ruscheweyh q-differential

operator to analytic functions of reciprocal order

Shahid Mahmood, Saima Mustafa and Imran Khan

Abstract. The core object of this paper is to define and study new class of analytic function using Ruscheweyhq-differential operator. We also investigate a number of useful properties such as inclusion relation, coefficient estimates, subordination result,for this newly subclass of analytic functions.

Mathematics Subject Classification (2010):30C45, 30C50.

Keywords:Analytic functions, Subordination, Functions with positive real part, Ruscheweyhq-differential operator, reciprocal order.

1. Introduction

Quantum calculus (q-calculus) is simply the study of classical calculus without the notion of limits. The study ofq-calculus attracted the researcher due to its applications in various branches of mathematics and physics, see detail [8]. Jackson [10, 12]

was the first to give some application ofq-calculus and introduced theq-analogue of derivative and integral. Later on Aral and Gupta [5, 6, 7] defined the q-Baskakov Durrmeyer operator by usingq-beta function while the author’s in [2, 3, 4] discussed theq-generalization of complex operators known asq-Picard andq-Gauss-Weierstrass singular integral operators. Recently, Kanas and R˘aducanu [13] definedq-analogue of Ruscheweyh differential operator using the concepts of convolution and then studied some of its properties. The application of this differential operator was further studied by Mohammed and Darus [1] and Mahmood and Sok´o l [14]. The aim of the current paper is to define a new class of analytic functions of reciprocal order involving q-differetial operator.

LetAbe the class of functions having the form f(z) =z+

∞

X

n=2

anzⁿ, (1.1)

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which are analytic in the open unit disk U= {z∈C:|z|<1}. LetM(α) denote a subclass ofAconsisting of functions which satisfy the inequality

Rezf⁰(z)

f(z) < α (z∈U),

for someα(α >1). And letN(α) be the subclass ofAconsisting of functionsf which satisfy the inequality:

Re(zf⁰(z))⁰

f⁰(z) < α (z∈U),

for someα(α >1). These classes were studied by Owa et al. [16, 18]. Shams et al. [20]

have introduced thek-uniformly starlikeSD(k, α) andk-uniformly convexCD(k, α) of order α,for somek(k≥0) andα(0≤α <1). Using these ideas in above defined classes, Junichi et al. [17] introduced the following classes.

Definition 1.1. Letf ∈ A. Thenf is said to be in classMD(k, α) if it satisfies Rezf⁰(z)

f(z) < k

zf⁰(z) f(z) −1

+α (z∈U), for someα(α >1) andk(k≤0).

Definition 1.2. An analytic functionf of the form (1.1) belongs to the classN D(k, α), if and only if

Re(zf⁰(z))⁰ f⁰(z) < k

(zf⁰(z))⁰ f⁰(z) −1

+α (z∈U), for someα(α >1) andk(k≤0).

If f and g are analytic in U, we say that f is subordinate to g, written as f ≺ g or f(z) ≺ g(z), if there exists a Schwarz function w, which is analytic in U with w(0) = 0 and|w(z)|<1 such that f(z) =g(w(z)). Furthermore, if the functiong(z) is univalent inU,then we have the following equivalence holds, see [11, 15].

f(z)≺g(z) (z∈U) ⇐⇒ f(0) =g(0) and f(U)⊂g(U).

For two analytic functions f(z) =

∞

P

n=1

anzⁿ g(z) =

∞

P

n=1

bnzⁿ (z∈U), Fort∈Randq >0,q6= 1, the number [t, q] is defined in [14] as

[t, q] = 1−q^t

1−q, [0, q] = 0.

For any non-negative integerntheq-number shift factorial is defined by [n, q]! = [1, q] [2, q] [3, q]· · ·[n, q], ([0, q]! = 1). We have lim

q→1[n, q] =n. Throughout in this paper we will assumeqto be fixed number between 0 and 1.

Theq-derivative operator orq-difference operator forf ∈ Ais defined as

∂qf(z) =f(qz)−f(z)

z(q−1) , z∈U.

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It can easily be seen that forn∈N:={1,2,3, . . .} andz∈U

∂qzⁿ= [n, q]zⁿ⁻¹, ∂q

( _∞ X

n=1

anzⁿ )

=

∞

X

n=1

[n, q]anzⁿ⁻¹. Theq-generalized Pochhammer symbol fort∈Randn∈Nis defined as

[t, q]_n= [t, q] [t+ 1, q] [t+ 2, q]· · ·[t+n−1, q], and fort >0, letq-gamma function is defined as

Γq(t+ 1) = [t, q] Γq(t) and Γq(1) = 1.

Definition 1.3. [14] For a function f(z)∈ A, the Ruscheweyhq-differential operator is defined as

D^µ_qf(z) =φ(q, µ+ 1;z)∗f(z) =z+

∞

X

n=2

Φn−1anzⁿ, (z∈Uandµ >−1), (1.2) where

φ(q, µ+ 1;z) =z+

∞

X

n=2

Φ_n−1zⁿ, (1.3)

and

Φ_n−1= Γq(µ+n)

[n−1, q]!Γq(µ+ 1) = [µ+ 1, q]_n−1

[n−1, q]! . (1.4)

From (1.2), it can be seen that

L⁰_qf(z) =f(z) andL¹_qf(z) =z∂_qf(z), and

L^m_q f(z) = z∂_q^m z^m−1f(z)

[m, q]! , (m∈N). lim

q→1⁻φ(q, µ+ 1;z) = z (1−z)^µ+1, and

lim

q→1⁻D^µ_qf(z) =f(z)∗ z (1−z)^µ+1.

This shows that in case of q →1⁻, the Ruscheweyh q-differential operator reduces to the Ruscheweyh differential operatorD^δ(f(z)) (see [19]). From (1.2) the following identity can easily be derived.

z∂D^µ_qf(z) =

1 + [µ, q]

q^µ

D^µ_qf(z)−[µ, q]

q^µ D^µ_qf(z). (1.5) Ifq→1⁻, then

z D^µ_qf(z)0

= (1 +µ)D^µ_qf(z)−µD^µ_qf(z).

Now using the Ruscheweyhq-differential operator, we define the following class.

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Definition 1.4. Letf ∈ A. Thenf is in the classKDq(k, α, γ) if Re

1 + 1

γ

z∂_qD^µ_qf(z) D^µqf(z) −1

< k 1 γ

+α, for somek(k≤0),α(α >1) and for someγ∈C\ {0}.

We note thatLD⁰₂(1,1, α) =M(α) andLD⁰₁(1,1, α) =N(α),the classes introduced by Owa et al. [16, 18]. When we takeγ= 1,2,c= 1, anda= 1 the classKDq(k, α, γ) reduces to the classesMD(k, α) andN D(k, α) (see [17]). For 1< α <4/3 the classes M(α) andN(α) were investigated by Uralegaddi et al. [21].

2. Preliminary results

Lemma 2.1. [9]For a positive integert, we have σ

t

X

j=1

(σ)_j−1

(j−1)! = (σ)_t

(t−1)!. (2.1)

Proof. Consider σ

t

X

j=1

(σ)_j−1 (j−1)!

= σ

1 +σ

1 +(σ)2

2! +(σ)3

3! +(σ)4

4! +· · ·+ (σ)t−1

(t−1)!

= σ(1 +σ)

1 +σ

2 +σ(σ+ 2)

2×3 +· · ·+σ(σ+ 2)· · ·(σ+t−2) 2× · · · ×(t−1)

= σ(1 +σ)(σ+ 2) 2

1 + σ

3 +· · ·+σ(σ+ 3)· · ·(σ+t−2) 3×4× · · · ×(t−1)

= σ(1 +σ)(σ+ 2) 2

(σ+ 3) 3

1 + σ

4 +· · ·+σ(σ+ 4)· · ·(σ+t−2) 4× · · · ×(t−1)

= σ(1 +σ)(σ+ 2) 2

(σ+ 3) 3

(σ+ 4) 4

1 + σ

5 +· · ·+ σ· · ·(σ+t−2) 5×6× · · · ×(t−1)

= σ(1 +σ)(σ+ 2) 2

(σ+ 3) 3

(σ+ 4) 4 · · ·

1 + σ

t−1

= σ(1 +σ)(σ+ 2) 2

(σ+ 3) 3

(σ+ 4) 4 · · ·

σ+ (t−1) t−1

= (σ)_t (t−1)!.

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3. Main results

With the help of the definition ofKDq(k, α, γ), we prove the following results.

Theorem 3.1. Iff(z)∈ KDq(k, α, γ), then f(z)∈ KDq

0,α−k 1−k, γ

.

Proof. Becausek≤0, we have Re

1 + 1

γ

z∂qD^µ_qf(z) D^µqf(z) −1

< k 1 γ

+α,

≤ kRe 1

γ

+α−k, which implies that

(1−k)Re1 γ

< α−k.

After simplification, we obtain Re

1 + 1

γ

<α−k

1−k,(k≤0, α >1 and ). (3.1)

This completes the proof.

Theorem 3.2. Iff(z)∈ KD_q(k, α, γ)and if f(z)has the form (1.1), then

|an| ≤ (σ)_n−1

(n−1)!Φ_n−1, (3.2)

where

σ=2|γ|(α−1)

q(1−k) . (3.3)

Proof. Let us define a function p(z) =

(α−k)−(1−k)h

1 +_γ¹_z∂_q_D^µ

qf(z) D^µ_qf(z) −1i

α−1 . (3.4)

Thenp(z) is analytic inU, p(0) = 1 andRe{p(z)}>0 forz∈U. We can write

1 + 1 γ

= (α−k)−(α−1)p(z)

1−k (3.5)

If we takep(z) = 1 +

∞

P

n=1

pnzⁿ, then (3.5) can be written as z∂_qD^µ_qf(z)−D^µ_qf(z) =−γ(α−1)

1−k D^µ_qf(z)

∞

X

n=1

p_nzⁿ

! . this implies that

"_∞ X

n=2

q[n−1] Φ_n−1anzⁿ

#

=−γ(α−1) 1−k

∞

X

n=1

Φ_n−1anzⁿ

! _∞ X

n=1

pnzⁿ

! .

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Using Cauchy product _∞

P

n=1

xn

· _∞

P

n=1

yn

=

∞

P

j=1 j

P

k=1

xkyk−j, we obtain

q[n−1] Φn−1anzⁿ =−γ(α−1) 1−k

∞

X

n=2





n−1

X

j=1

Φj−1ajpn−j



zⁿ.

Comparing the coefficients ofnthterm on both sides, we obtain an = −γ(α−1)

q[n−1] Φ_n−1(1−k)

n−1

X

j=1

Φj−1ajpn−j. By taking absolute value and applying triangle inequality, we get

|an| ≤ |γ|(α−1) q[n−1] Φn−1(1−k)

n−1

X

j=1

Φ_j−1|aj| |pn−j|.

Applying the coefficient estimates|pn| ≤2 (n≥1) for Caratheodory functions [11], we obtain

|a_n| ≤ 2|γ|(α−1) q[n−1] Φ_n−1(1−k)

n−1

X

j=1

Φ_j−1|a_j|

= σ

[n−1] Φn−1 n−1

X

j=1

ψ_j−1|a_j|, (3.6)

whereσ= 2|γ|(α−1)/q(1−k). To prove (3.2) we apply mathematical induction. So forn= 2, we have from (3.6)

|a2| ≤ σ Φ1

= (σ)₂₋₁

[2−1]!Φ₂₋₁, (3.7)

which shows that (3.2) holds forn= 2. Forn= 3, we have from (3.6)

|a3| ≤ σ

[3−1] Φ₃₋₁{1 + Φ1|a2|}, using (3.7), we have

|a3| ≤ σ

[2] Φ₂(1 +σ) = (σ)₃₋₁ [3−1] Φ₃₋₁,

which shows that (3.2) holds for n= 3. Let us assume that (3.2) is true for n≤t, that is,

|at| ≤ (σ)_t−1

[t−1]!Φ_t−1 j= 1,2, . . . , t. (3.8)

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Using (3.6) and (3.8), we have

|at+1| ≤ σ tΦt

t

X

j=1

Φj−1|aj|

≤ σ

tΦt t

X

j=1

ψ_j−1 (σ)_j−1 [j−1]!Φj−1

= σ

tΦ_t

t

X

j=1

(σ)_j−1 [j−1]!. Applying (2.1), we have

|at+1| ≤ 1 tΦ_t

(σ)t

[t−1]!

= 1

Φt

(σ)_t [t]! .

Consequently, using mathematical induction, we have proved that (3.2) holds true for

alln,n≥2. This completes the proof.

Theorem 3.3. If a functionf ∈ KDq(k, α, γ), then z∂_qD^µ_qf(z)

D^µqf(z) ≺1 + 2 (α1−1)−2 (α1−1)

1−z (z∈U), (3.9)

α₁=α−k

1−k. (3.10)

Proof. Iff(z)∈ KDq(k, α, γ), then by (3.1) Re

1 + 1

γ

< α1. (3.11)

Then there exists a Schwarz functionw(z) such that α1−n

1 +_γ¹_z∂_q_D^µ

qf(z) D^µ_qf(z) −1o

α₁−1 =1 +w(z)

1−w(z), (3.12)

and

Re

1 +w(z) 1−w(z)

>0, (z∈U).

Therefore, from (3.12), we obtain z∂qD^µ_qf(z)

D^µqf(z) = 1 +γ(α1−1)

1−1 +w(z) 1−w(z)

. This gives

z∂_qD^µ_qf(z)

D^µqf(z) = 1 + 2γ(α1−1)−2γ(α1−1) 1−w(z)

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and hence

z∂qD^µ_qf(z)

D^µqf(z) ≺1 + 2γ(α1−1)−2γ(α1−1)

1−z (z∈U).

which was required in (3.9).

Theorem 3.4. If functionf ∈ KDq(k, α, γ), then we have 1−[1 + 2γ(α1−1)]r

1−r ≤Re

z∂_qD^µ_qf(z) D^µqf(z)

≤ 1 + [1 + 2γ(α1−1)]r

1 +r , (3.13)

for|z|=r <1 andα1 is defined by (3.10).

Proof. By the virtue of Theorem (3.3), let us take the functionφ(z) defined by φ(z) = 1 + 2γ(α₁−1)−2γ(α₁−1)

1−z (z∈U). Lettingz=re^iθ(0≤r <1),we see that

Reφ(z) = 1 + 2γ(α1−1) + 2γ(1−α₁) (1−rcosθ) 1 +r²−2rcosθ . Let us define

ψ(t) = 1−rt

1 +r²−2rt (t= cosθ). Sinceψ⁰(t) = r 1−r²

(1 +r²−2rt)² ≥0,becauser <1. Therefore we get 1 + 2γ(α1−1)−2γ(α₁−1)

1−r ≤Reφ(z)≤1 + 2γ(α1−1)−2γ(α₁−1) 1 +r . After simplification, we have

1−[1 + 2γ(α₁−1)]r

1−r ≤Reφ(z)≤1 + [1 + 2γ(α₁−1))]r

1 +r .

Since we note that z∂qD^µ_qf(z)

D^µqf(z) ≺φ(z),(z∈U) by Theorem 3.3 andφ(z) is analytic

inU,we proved the inequality (3.13).

Theorem 3.5. Iff ∈ A satisfies

< (α−1)|γ|

(1−k) z∈U, (3.14)

for somek(k≤0),α(α >1) andγ∈C\ {0}. Thenf ∈ KDq(k, α, γ).

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Proof.

<(α−1)|γ|

(1−k)

⇒ 1 γ

< α−1 1−k

⇒ (1−k) 1 γ

+ 1< α

⇒ 1 γ

+ 1< k 1 γ

+α

⇒ Re

1 + 1 γ

+ 1< k 1 γ

+α

⇒ f ∈ LD^k_b(a, c, β)

Corollary 3.6. Let f ∈ Abe of the form (1.1)and satisfies

P∞

n=2[n−1] Φ_n−1a_nzⁿ⁻¹ 1 +P∞

n=2Φ_n−1anzⁿ⁻¹

< (α−1)|γ|

q(1−k) z∈U, (3.15) for somek(k≤0),β(β >1)and for someb∈C\ {0}. Thenf ∈ KD_q(k, α, γ)..

Proof. We have

D^µ_qf(z) =z+

∞

X

n=2

Φ_n−1a_nzⁿ and by (1.5)

z∂D^µ_qf(z) =z+

∞

X

n=2

[n] Φ_n−1a_nzⁿ.

Therefore, (3.14) follows immediately (3.15).

Theorem 3.7. Letf ∈ Abe of the form (1.1)and satisfies

∞

X

n=2

([n−1] +y)|Φ_n−1||an|< y z∈U, (3.16) for somek(k≤0),β(β >1)and for someb∈C\ {0} and where

y= (α−1)|γ|

q(1−k) >0.

Thenf ∈ KDq(k, α, γ).

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Proof. We have

∞

X

n=2

([n−1] +y)|Φ_n−1||an|< y

⇒

∞

X

n=2

([n−1] +y)|Φn−1||an|< y−y

∞

X

n=2

|Φn−1||an|

⇒ 0< y−y

∞

X

n=2

|Φ_n−1||an|

⇒ 0< y−y

∞

X

n=2

|Φn−1||an||zⁿ⁻¹|

⇒ 0< y

1 +

∞

X

n=2

Φ_n−1anzⁿ⁻¹

(3.17) We have

∞

X

n=2

([n−1] +y)|Φ_n−1||an|< y

⇒

∞

X

n=2

([n−1] +y)|Φn−1||an||zⁿ⁻¹|< y

⇒

∞

X

n=2

[n−1]|Φ_n−1||a_n||zⁿ⁻¹|< y−y

∞

X

n=2

|Φ_n−1||a_n||zⁿ⁻¹|

⇒

∞

X

n=2

[n−1] Φ_n−1a_nzⁿ⁻¹

< y

1 +

∞

X

n=2

Φ_n−1a_nzⁿ⁻¹

⇒

P∞

n=2[n−1] Φ_n−1a_nzⁿ⁻¹ 1 +P∞

n=2Φ_n−1a_nzⁿ⁻¹

< y,

because of (3.17). By (3.15) it followsf ∈ LD^k_b(a, c, β).

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Shahid Mahmood Corresponding author

Department of Mechanical Engineering, Sarhad University of Science and

I.T. Landi Akhun Ahmad, Hayatabad Link. Ring Road, Peshawar, Pakistan e-mail:[email protected]

Saima Mustafa

Department of Statistics & Mathematics PMAS-Arid Agriculture University, Rawalpindi e-mail:[email protected]

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Imran Khan

Department of Basic Sciences and Islamyat University of Engineering and Technology Peshawar, Pakistan

e-mail:[email protected]