Properties of a New Subclass of Analytic Functions With Negative Coefficients Defined by Using the Q-Derivative

Roberta Bucur 1  and Daniel Breaz 2
  • 1 Department of Mathematics, Pitesti, Romania
  • 2 Department of Mathematics, Bucuresti, Romania
Roberta Bucur and Daniel Breaz

Abstract

In this paper we define a new class of analytic functions with negative coefficients involving the q-differential operator. Our main purpose is to determine coefficient inequalities and distortion theorems for functions belonging to this class. Connections with previous results are pointed out.

1 Introduction

The quantum calculus or q–calculus has called the attention of many great researchers from Geometric function theory field due to its numerous applications in mathematics and physics. In 1999 was defined the first q-analogue of a starlike function by Ismail et al. [1]. In fact, many results obtained for univalent functions can be extended by using q-analogues functions. Recently, analytic functions with negative coefficient were studied in papers [5, 8, 9, 10].

Let U = {z : |z| < 1} be the open unit disk of the complex plane and let 𝒜 represent the class of all functions of the form

f(z)=z+n=2anzn,zU.

For 0 < q < 1, the q-derivative of a function f ∈ 𝒜 is defined by (see [2])

dqf(z)=f(qz)f(z)(q1)z,(z0),
with dq f (0) = f′ (0).

Motivated by the aforementionated works, we define the following class of functions associated with Janowski functions:

Definition 1.1

For 0 ≤ μ ≤ 1, k ≥ 0 and −1 ≤ B < A ≤ 1, we let ST (q, μ,k;A,B) be the subclass of 𝒜 consisting of functions of the form (1.1) and satisfying the condition

Re{(B1)zdqf(z)/[(1μ)z+μf(z)](A1)(B+1)zdqf(z)/[(1μ)z+μf(z)](A+1)}>
k|(B1)zdqf(z)/[(1μ)z+μf(z)](A1)(B+1)zdqf(z)/[(1μ)z+μf(z)](A+1)1|,zU.

Let 𝒯 denote the subclass of analytic functions f ∈ 𝒜 of the form

f(z)=zn=2anzn,an0.
Further, let 𝒯ST (q, μ,k;A,B) ≡ ST (q, μ,k;A,B) ∩ 𝒯.

Also, we remark that ST (q, μ,k;A,B) reduces to the following known classes:

  1. (i)In case A = 1 − 2α, 0 ≤ α < 1, B = −1, μ = 1 and q → 1, we obtain the class Sp(k,α) of k–uniformly starlike functions of order α(see [4] );
  2. (ii)In case A = 1, B = −1, μ = 1, k = 1 and q → 1, we get the class Sp of uniformly starlike functions (see [6]);
  3. (iii)In case μ = 1 and q → 1, we obtain the class kST [A,B] which was introduced and studied by Noor and Malik (see [3]);
  4. (iv)In case A = 1 − 2α, 0 ≤ α < 1, B = −1, μ = 1, k = 0 and q → 1, we get the class of starlike functions of order α denoted S* (α);
  5. (v)In case A = 1, B = −1, μ = 0, k = 0 and q → 1, we obtain the class R of functions whose derivative has positive real part.

Unless otherwise mentioned, we assume throughout this paper that 0 ≤ μ ≤ 1, k ≥ 0 and −1 ≤ B < A ≤ 1.

2 Coefficient Estimates

We begin with a result that provides coefficient inequalities for functions in the class ST (q, μ,k;A,B).

Theorem 2.1

A function f ∈ 𝒜 of the form (1.1) is in the class ST (q, μ,k;A,B) if

n=2{2(k+1)([n]qμ)+|[n]q(B+1)μ(A+1)|}|an||BA|.

Proof

It is suffices to prove that

k|g(z)1|Re[g(z)1]<1,zU,
where the function g is defined by
g(z):=(B1)zdqf(z)/[(1μ)z+μf(z)](A1)(B+1)zdqf(z)/[(1μ)z+μf(z)](A+1),zU.

First of all,

k|g(z)1|Re[g(z)1](k+1)|g(z)1|,zU.

Because

(k+1)|g(z)1|=2(k+1)|n=2(μ[n]q)anzn1||(BA)+n=2{(B+1)[n]qμ(A+1)}anzn1|2(k+1)n=2(μ[n]q)|an||BA|n=2{(B+1)[n]qμ(A+1)}|an|
and (2.1) take place, then (k + 1)|g(z) − 1| is bounded above by 1. Hence fST (q, μ,k;A,B) and the theorem is proved.

Corollary 2.1

(see [3], Theorem 2.1) Let function f ∈ 𝒜 be of the form (1.1). If

n=2{2(k+1)(n1)+|n(B+1)(A+1)|}|an||BA|,
then fkST [A,B].

Corollary 2.2

(see [7], Theorem 1) Let function f ∈ 𝒜 be of the form (1.1). If

n=2(nα)|an|1α,
then function f is in the class S* (α).

For f ∈ 𝒯ST (q, μ,k;A,B) the converse of Theorem 2.1 is also true.

Theorem 2.2

A function f ∈ 𝒯 given by (1.5) is in the class 𝒯ST (q, μ,k;A,B) if and only if

n=2{2(k+1)([n]qμ)+|[n]q(B+1)μ(A+1)|}an|BA|.
The result is sharp with the extremal function f given by
f(z)=z|BA|2(k+1)([n]qμ)+|[n]q(B+1)μ(A+1)|zn,zU.

Proof

In view of Theorem 2.1, we need only to prove that (2.8) holds if f ∈ 𝒯ST (q, μ,k;A,B). Conversely, assuming that f ∈ 𝒯ST (q, μ,k;A,B) and z is real, we find that g(z) ≥ k|g(z) − 1|, where g is given in (2.3). Therefore,

(BA)n=2{(B1)[n]qμ(A1)}anzn1(BA)n=2{(B+1)[n]qμ(A+1)}anzn12k|n=2([n]qμ)anzn1(BA)n=2{(B+1)[n]qμ(A+1)}anzn1|.
Next, letting z → 1 through real values, we obtain the inequality (2.8). Also, the equality in (2.8) take place for the function f given in (2.9).

Corollary 2.3

Let function f be of the form (1.5). Then fkST [A,B] ∩ 𝒯 if and only if

n=2{2(k+1)(n1)+|n(B+1)(A+1)|}an|BA|.

Corollary 2.4

(see [7], Theorem 2) Let function f ∈ 𝒯 be of the form (1.5). Then f is a starlike function of order α if and only if

n=2(nα)an1α.

3 Distortion Theorems

Theorem 3.1

Let the function f of the form (1.5) be in the class 𝒯ST (q, μ,k;A,B). Then, for |z| = r,

|f(z)|r+|BA|2(k+1)([2]qμ)+|[2]q(B+1)μ(A+1)|r2
and
|f(z)|r|BA|2(k+1)([2]qμ)+|[2]q(B+1)μ(A+1)|r2,
with equality for
f(z)=z|BA|2(k+1)([2]qμ)+|[2]q(B+1)μ(A+1)|z2.

Proof

By using the hypothesis and the triangle inequality, we find that

rr2n=2an|f(z)|r+r2n=2an,
which, in conjunction with (2.8) gives us the inequalities (3.1) and (3.2).

Corollary 3.1

Let the function f given by (1.5) be in the class kST [A,B] ∩ 𝒯. Then, for |z| = r,

r|BA|2(k+1)+|2BA+1|r2|f(z)|r+|BA|2(k+1)+|2BA+1|r2,
with equality for
f(z)=z|BA|2(k+1)+|2BA+1|z2.

Corollary 3.2

(see [7], Theorem 4) Let the function f ∈ 𝒯 given by (1.5) be in the class S* (α) ∩ 𝒯. Then, for |z| = r,

r1α2αr2|f(z)|r+1α2αr2,
with equality for
f(z)=z1α2αz2.

The next theorem can be proven by employing similar techniques as in the demonstration of Theorem 3.1, so we will omit the details of our proof.

Theorem 3.2

Let the function f of the form (1.5) be in the class 𝒯ST (q, μ,k;A,B). Then, for |z| = r,

|f(z)|1+2|BA|2(k+1)([2]qμ)+|[2]q(B+1)μ(A+1)|r
and
|f(z)|12|BA|2(k+1)([2]qμ)+|[2]q(B+1)μ(A+1)|r,
with equality for f given by (3.3).

Corollary 3.3

Let the function f given by (1.5) be in the class kST [A,B] ∩ 𝒯. Then, for |z| = r,

12|BA|2(k+1)+|2BA+1|r|f(z)|1+2|BA|2(k+1)+|2BA+1|r,
with equality for f given by (3.5).

Corollary 3.4

(see [7], Theorem 4) Let the function f ∈ 𝒯 given by (1.5) be in the class S* (α) ∩ 𝒯. Then, for |z| = r,

12(1α)2α)r|f(z)|1+2(1α)2αr,
with equality for f given by (3.6).

References

  • [1]

    M.E.H Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Var. Theory Appl., vol. 14, 77–84, 1999.

  • [2]

    F.H. Jackson, On q-functions and certain difference operator, Trans. R. Soc. Edinburgh, vol. 46, 253–281, 1908.

  • [3]

    K. L. Noor and S. N. Malik, On coefficient inequalities of functions associatd with conic domains, Comput. Math. Appl. 62, 2209–2219, 2011.

  • [4]

    S. Owa, On uniformly convex functions, Math. Japon, 48, 377–384, 1998.

  • [5]

    Akhter Rasheed, Saqib Hussain, Muhammad Asad Zaighum and Maslina Darus, Class of Analytic functions related with uniformly convex and Janowski functions, Journal of Function Spaces, Vol. 2018, Article ID 4679857, 1–6, 2018.

  • [6]

    F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., vol. 18, 189–196, 1993.

  • [7]

    Herb Silverman, Univalent functions with negative coefficients, Proceedings of the American Mathematical Society, vol. 51, no. 1, 1975.

  • [8]

    H.M. Srivastava, R.M. El-Ashwah, N. Breaz, A certain Subclass of Multivalent Functions Involving Higher-Order Derivatives, Filomat, vol. 30, no.1, 113–124, 2016.

  • [9]

    H.M. Srivastava, M. Tahir, B. Khan, Q.Z. Ahmad and N. Khan, Some General Classes of q-Starlike Functions Associated with the Janowski Functions, Symmetry, vol. 292, no.11, 1–14, 2019.

  • [10]

    K. Vijaya, G. Murugusundaramoorthy, S. Yalcin, Certain Class of Analytic Functions Involving Salagean Type q-Difference Operator, Konuralp Journal of Mathematics, vol.6, no.2, 264–271, 2018.

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  • [1]

    M.E.H Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Var. Theory Appl., vol. 14, 77–84, 1999.

  • [2]

    F.H. Jackson, On q-functions and certain difference operator, Trans. R. Soc. Edinburgh, vol. 46, 253–281, 1908.

  • [3]

    K. L. Noor and S. N. Malik, On coefficient inequalities of functions associatd with conic domains, Comput. Math. Appl. 62, 2209–2219, 2011.

  • [4]

    S. Owa, On uniformly convex functions, Math. Japon, 48, 377–384, 1998.

  • [5]

    Akhter Rasheed, Saqib Hussain, Muhammad Asad Zaighum and Maslina Darus, Class of Analytic functions related with uniformly convex and Janowski functions, Journal of Function Spaces, Vol. 2018, Article ID 4679857, 1–6, 2018.

  • [6]

    F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., vol. 18, 189–196, 1993.

  • [7]

    Herb Silverman, Univalent functions with negative coefficients, Proceedings of the American Mathematical Society, vol. 51, no. 1, 1975.

  • [8]

    H.M. Srivastava, R.M. El-Ashwah, N. Breaz, A certain Subclass of Multivalent Functions Involving Higher-Order Derivatives, Filomat, vol. 30, no.1, 113–124, 2016.

  • [9]

    H.M. Srivastava, M. Tahir, B. Khan, Q.Z. Ahmad and N. Khan, Some General Classes of q-Starlike Functions Associated with the Janowski Functions, Symmetry, vol. 292, no.11, 1–14, 2019.

  • [10]

    K. Vijaya, G. Murugusundaramoorthy, S. Yalcin, Certain Class of Analytic Functions Involving Salagean Type q-Difference Operator, Konuralp Journal of Mathematics, vol.6, no.2, 264–271, 2018.

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