Research article

On deep holes of generalized Reed-Solomon codes

  • Received: 16 May 2016 Accepted: 20 June 2016 Published: 28 June 2016
  • Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word u is a deep hole of the standard Reed-Solomon codes [q-1, k]q if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.

    Citation: Shaofang Hong, Rongjun Wu. On deep holes of generalized Reed-Solomon codes[J]. AIMS Mathematics, 2016, 1(2): 96-101. doi: 10.3934/Math.2016.2.96

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  • Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word u is a deep hole of the standard Reed-Solomon codes [q-1, k]q if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.


    Let A denote the class of functions of the form

    f(z)=z+k=2akzk, (1.1)

    which are analytic in the open disc U={zC:|z|<1}. Let S denote the subclass of A consisting of functions that are univalent in U. Also, let Ω be the class of all analytic functions w in U that satisfy the conditions w(0)=0 and |w(z)|<1(zU). If f and g are analytic in U, we say that f is subordinate to g, written as fg in U or f(z)g(z) (zU), if there exists wΩ such that f(z)=g(w(z)) (zU). Furthermore, if the function g(z) is univalent in U, then we have the following equivalence holds (see [4] and [11]):

    f(z)g(z)f(0)=g(0)  and  f(U)g(U).

    A function fA is said to be in the class of γ spiral-like functions of order λ in U, denoted by S(γ;λ) if

    {eiγzf(z)f(z)}>λ cosγ   (0λ<1,|γ|<π2;zU). (1.2)

    The class S(γ;λ) was studied by Libera [10] and Keogh and Merkes [9]. Note that

    1). S(γ;0)=S(γ) is the class of spiral-like functions introduced by Špaček [17];

    2). S(0;λ)=S(λ) is the class of starlike functions of order λ;

    3). S(0;0)=S is the familiar class of starlike functions.

    For functions fA given by (1.1) and gA given by

    g(z)=z+k=2bkzk, (1.3)

    we define the Hadamard product (or Convolution) of f and g by

    (fg)(z)=z+k=2akbkzk. (1.4)

    Also, for fA given by (1.1) and 0<q<1, the Jackson's q-derivative operator or q-difference operator for a function fA is defined by (see [1,2,3,6,7,15,16])

    Dqf(z):={f(0)if z=0,f(z)f(qz)(1q)zif z0. (1.5)

    From (1.5), we deduce that

    Dqf(z)=1+k=2[k]q akzk1 (z0), (1.6)

    where the q-integer number [i]q is defined by

    [i]q=1qi1q=1+q+q2+...+qi1, (1.7)

    and

    limq1Dqf(z)=limq1f(z)f(qz)(1q)z=f(z), (1.8)

    for a function f which is differentiable in a given subset of C.

    Next, in terms of the q-generalized Pochhammer symbol ([v]q)n given by

    ([v]q)n=[v]q [v+1]q [v+2]q ... [v+n1]q, (1.9)

    we define the function ϕq(a,c;z) by

    ϕq(a,c;z)=z+k=2([a]q)k1([c]q)k1zk   (aR;cRZ0;Z0={0,1,2,...};zU). (1.10)

    Corresponding to the function ϕq(a,c;z), we consider a linear operator Lq(a,c):AA which is defined by means of the following Hadamard product (or convolution):

    Lq(a,c)f(z)=ϕq(a,c;z)f(z)=z+k=2([a]q)k1([c]q)k1 ak zk. (1.11)

    It is easily verified from (1.11) that

    qa1zDq(Lq(a,c)f(z))=[a]q Lq(a+1,c)f(z)[a1]qLq(a,c)f(z). (1.12)

    Moreover, for fA, we observe that

    1). limq1Lq(a,c)f(z)=L(a,c)f(z), where L(a,c) denotes the Carlson-Shaffer operator [5];

    2). Lq(δ+1,1)f(z)=Rδqf(z)(δ>0), where Rδq denotes the Ruscheweyh q-derivative of a function fA of order δ (see [8]);

    3). limq1Lq(δ+1,1)f(z)=Rδf(z)(δ>0), where Rδq denotes the Ruscheweyh derivative of order δ (see [14]);

    4). Lq(a,a)f(z)=f(z) and Lq(2,1)f(z)=zDqf(z).

    Making use of the q-analogue of Carlson-Shaffer operator Lq(a,c), we introduce a new subclass of spiral-like functions.

    Definition 1. For 0t1, 0λ<1 and |γ|<π2, we let Saq(γ,λ,t) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:

    {eiγzDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)}>λ cosγ (1.13)
    (0t1;0λ<1;|γ|<π2;zU).

    We note that

    1). For t=1, 0λ<1 and |γ|<π2, we let Saq(γ,λ,1)=Saq(γ,λ) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:

    {eiγzDq(Lq(a,c)f(z))Lq(a,c)f(z)}>λ cosγ (1.14)
    (0λ<1;|γ|<π2;zU).

    2). For t=0, 0λ<1 and |γ|<π2, we let Saq(γ,λ,0)=Kaq(γ,λ) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic condition:

    {eiγ Dq(Lq(a,c)f(z))}>λ cosγ (1.15)
    (0λ<1;|γ|<π2;zU).

    The object of the present paper is to investigate the coefficient estimates and subordination properties for the class of functions Saq(γ,λ,t). Some interesting consequences of the results are also pointed out.

    In this section, we obtain several sufficient conditions for a function fA to be in the class Saq(γ,λ,t).

    Theorem 1. Let fA and let σ be a real number with 0σ<1. If

    |zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)1|1σ   (zU), (2.1)

    then fSaq(γ,λ,t) provided that

    |γ|cos1(1σ1λ) (2.2)

    Proof. From (2.1) it follows that

    zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)=1+(1σ)w(z),

    where w(z)Ω. We have

    {eiγzDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)}={eiγ[1+(1σ)w(z)]}=cosγ+(1σ){eiγw(z)}cosγ(1σ)|eiγw(z)|>cosγ(1σ)λ cosγ

    provided that |γ|cos1(1σ1λ). Thus, the proof is completed.

    Putting σ=1(1λ)cosγ in Theorem 1, we obtain the following result.

    Corollary 1. Let fA. If

    |zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)1|(1λ)cosγ   (zU), (2.3)

    then fSaq(γ,λ,t).

    In the following theorem, we obtain a sufficient condition for f to be in Saq(γ,λ,t).

    Theorem 2. A function f(z) of the form (1.1) is in Saq(γ,λ,t) if

    k=2{([k]qt)secγ+(1λ)t}([a]q)k1([c]q)k1|ak|1λ. (2.4)

    Proof. In virtue of Corollary 1, it suffices to show that the condition (2.3) is satisfied. We have

    |zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)1|=|k=2{[k]qt}([a]q)k1([c]q)k1 ak zk11+tk=2([a]q)k1([c]q)k1ak zk1|<k=2{[k]qt}([a]q)k1([c]q)k1 |ak|1k=2t([a]q)k1([c]q)k1|ak|.

    The last expression is bounded above by (1λ)cosγ, if

    k=2{[k]qt}([a]q)k1([c]q)k1 |ak|(1λ)cosγ{1k=2t([a]q)k1([c]q)k1|ak|}

    which is equivalent to

    k=2{([k]qt)secγ+(1λ)t}([a]q)k1([c]q)k1|ak|1λ.

    This completes the proof of the Theorem 2.

    Putting t=1 in Theorem 2, we obtain the following corollary.

    Corollary 2. A function f(z) of the form (2.1) is in Saq(γ,λ) if

    k=2{([k]q1)secγ+1λ}([a]q)k1([c]q)k1|ak|1λ. (2.5)

    Putting t=0 in Theorem 2, we obtain the following corollary.

    Corollary 3. A function f(z) of the form (2.1) is in Kaq(γ,λ) if

    k=2[k]qsecγ([a]q)k1([c]q)k1|ak|1λ. (2.6)

    Before stating and proving our subordination result for the class Saq(γ,λ,t), we need the following definitions and a lemma due to Wilf [19].

    Definition 2 [19]. A sequence {bk}k=1 of complex numbers is said to be a subordinating factor sequence if, whenever f(z)=z+k=2akzk is regular, univalent and convex in U, we have

    k=1ak bk f(z)   (a1=1;zU). (3.1)

    Lemma 1 [19]. The sequence {bk}k=1 is a subordinating factor sequence if and only if

    {1+2k=1bk zk}>0   (zU). (3.2)

    Theorem 3. Let fSaq(γ,λ,t) satisfy the coefficient inequality (2.4) with ac>0 and let g(z) be any function in the usual class of convex functions C, then

    {([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q](fg)(z)g(z) (3.3)

    and

    {f(z)}>1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q{([2]qt)secγ+(1λ)t}[a]q[c]q. (3.4)

    The constant factor {([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q] in (3.3) cannot be replaced by a larger number.

    Proof. Let fSaq(γ,λ,t) satisfy the coefficient inequality (2.4) and suppose that

    g(z)=z+k=2bkzkC.

    Then, by Definition 2, the subordination (3.3) of our theorem will hold true if the sequence

    {{([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q]ak}k=1

    is a subordinating factor sequence, with b1=1. In view of Lemma 1, it is equivalent to the inequality

    {1+k=1{([2]qt)secγ+(1λ)t}[a]q[c]q1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qak zk}>0   (zU). (3.5)

    By noting the fact that {([k]qt)secγ+(1λ)t}([a]q)k1(1λ)([c]q)k1 is an increasing function for k2 and ac>0. In view of (2.4), when |z|=r<1, we obtain

    {1+{([2]qt)secγ+(1λ)t}[a]q[c]q1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qk=1ak zk}
    ={1+{([2]qt)secγ+(1λ)t}[a]q[c]q1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qz+k=2{([2]qt)secγ+(1λ)t}[a]q[c]qak zk1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q}
    1{([2]qt)secγ+(1λ)t}[a]q[c]q1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qrk=2{([k]qt)secγ+(1λ)t}([a]q)k1([c]q)k1|ak| rk1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q
    1{([2]qt)secγ+(1λ)t}[a]q[c]q1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qr1λ1λ+{([2]qt)secγ+(1λ)t}[a]q[c]qr=1r>0   (|z|=r<1).

    This evidently proves the inequality (3.5) and hence also the subordination result (3.3) asserted by Theorem 3. The inequality (3.4) follows from (3.3) by taking

    g(z)=z1z=z+k=2zkC.

    The sharpness of the multiplying factor in (3.3) can be established by considering a function

    F(z)=z1λ{([2]qt)secγ+(1λ)t}[a]q[c]qz2.

    Clearly FSaq(γ,λ,t) satisfy (2.4). Using (3.3) we infer that

    {([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q]F(z)z1z,

    and it follows that

    min|z|r{{([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q]{F(z)}}=12.

    This shows that the constant {([2]qt)secγ+(1λ)t}[a]q[c]q2[1λ+{([2]qt)secγ+(1λ)t}[a]q[c]q] cannot be replaced by any larger one.

    For t=1 in Theorem 3, we state the following corollary.

    Corollary 4. Let fSaq(γ,λ) satisfy the coefficient inequality (2.5) with ac>0 and gC, then

    {qsecγ+1λ}[a]q[c]q2[1λ+{qsecγ+1λ}[a]q[c]q](fg)(z)g(z) (3.6)

    and

    {f(z)}>1λ+{qsecγ+1λ}[a]q[c]q{qsecγ+1λ}[a]q[c]q. (3.7)

    The constant factor {qsecγ+1λ}[a]q[c]q2[1λ+{qsecγ+(1λ)}[a]q[c]q] in (3.6) cannot be replaced by a larger number.

    Taking t=0 in Theorem 3, we state the next corollary.

    Corollary 5. Let fKaq(γ,λ) satisfy the coefficient inequality (2.6) with ac>0 and gC, then

    (1+q)secγ[a]q[c]q2[1λ+(1+q)secγ[a]q[c]q](fg)(z)g(z) (3.8)

    and

    {f(z)}>1λ+(1+q)secγ[a]q[c]q(1+q)secγ[a]q[c]q. (3.9)

    The constant factor (1+q)secγ[a]q[c]q2[1λ+(1+q)secγ[a]q[c]q] in (3.8) cannot be replaced by a larger number.

    The Fekete-Szegö problem consists in finding sharp upper bounds for the functional |a3μa22| for various subclasses of A (see [13] and [18]). In order to obtain sharp upper-bounds for |a3μa22| for the class Saq(γ,λ,t) the following lemma is required (see, e.g., [12, p.108]).

    Lemma 2. Let the function wΩ be given by

    w(z)=k=1wk zk   (zU).

    Then

    |w1|1,   |w2|1|w1|2, (4.1)

    and

    |w2s w21|max{1,|s|}, (4.2)

    for any complex number s. The functions w(z)=z and w(z)=z2or one of their rotations show that both inequalities (4.1) and (4.2) are sharp.

    For the constants λ, γ with 0λ<1 and |γ|<π2 denote

    Pλ,γ(z)=1+eiγ(eiγ2λcosγ)z1z   (zU). (4.3)

    The function Pλ,γ(z) maps the open unit disk U onto the half-plane Hλ,γ={wC:{eiγw}>λcosγ}. If

    Pλ,γ(z)=1+k=1pkzk   (zU),

    then it is easy to check that

    pk=2eiγ(1λ)cosγ    (k1). (4.4)

    First we obtain sharp upper-bounds for the Fekete-Szegö functional |a3μa22| with μ real parameter.

    Theorem 4. Let fSaq(γ,λ,t) be given by (1.1) and let μ be a real number. Then

    |a3μa22|{2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[1+2(1λ)t1+qt μ2(1λ)(1+q+q2t)([a]q)2([c]q)2(1+qt )2([a]q)2([c]q)2](μσ1)2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2(σ1μσ2)2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[12(1λ)t1+qt +μ2(1λ)(1+q+q2t)([a]q)2([c]q)2(1+qt )2([a]q)2([c]q)2](μσ2) (4.5)

    where

    σ1=t(1+qt) ([a]q)2([c]q)2(1+q+q2t)([c]q)2([a]q)2 (4.6)
    σ2= (1+qt)(1+qtλ)([a]q)2([c]q)2(1λ)(1+q+q2t)([a]q)2 ([c]q)2 (4.7)

    and all estimates are sharp.

    Proof. Suppose that fSaq(γ,λ,t) is given by (1.1). Then, from the definition of the class Saq(γ,λ,t), there exists wΩ,

    w(z)=w1z+w2z2+w3z3+...

    such that

    zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)=Pλ,γ(w(z))   (zU). (4.8)

    We have

    zDq(Lq(a,c)f(z))(1t) z+t Lq(a,c)f(z)=1+(1+qt)[a]q[c]qa2 z
    +{(1+q+q2t)([a]q)2([c]q)2 a3t(1+qt)([a]q)2([c]q)2 a22}z2+... (4.9)

    Set

    Pλ,γ(z)=1+p1z+p2z2+p3z3+... . 

    From (4.4) we have

    p1=p2=2eiγ(1λ)cosγ.

    Equating the coefficients of z and z2 on both sides of (4.8) and using (4.9), we obtain

    a2=p1 [c]q(1+qt) [a]qw1

    and

    a3=([c]q)2(1+q+q2t)([a]q)2[p1w2+(p2+t (1+qt) p21)w21]

    and thus we obtain

    a2=2eiγ(1λ)cosγ [c]q(1+qt) [a]qw1 (4.10)

    and

    a3=2eiγ(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[w2+(1+2teiγ(1λ)cosγ 1+qt )w21]. (4.11)

    It follows

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[|w2|+|1+2eiγ(1λ)cosγ1+qt (t μ(1+q+q2t)([a]q)2([c]q)2(1+qt) ([a]q)2([c]q)2)||w1|2]

    Making use of Lemma 2 we have

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[1+(|1+2eiγ(1λ)cosγ1+qt (t μ(1+q+q2t)([a]q)2 ([c]q)2([c]q)2(1+qt) ([a]q)2)|1)|w1|2]

    or

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[1+(1+M(M+2)cos2γ1)|w1|2], (4.12)

    where

    M=2(1λ)1+qt (t μ(1+q+q2t)([a]q)2 ([c]q)2([c]q)2(1+qt) ([a]q)2). (4.13)

    Denote by

    F(x,y)=[1+(1+M(M+2)x21)y2]

    where x=cosγ, y=|w1| and (x,y):[0,1]×[0,1].

    Simple calculation shows that the function F(x,y) does not have a local maximum at any interior point of the open rectangle (0,1)×(0,1). Thus, the maximum must be attained at a boundary point. Since F(x,0)=1, F(0,y)=1 and F(1,1)=|1+M|, it follows that the maximal value of F(x,y) may be F(0,0)=1 or F(1,1)=|1+M|. Therefore, from (4.12) we obtain

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2max{1,|1+M|}, (4.14)

    where M is given by (4.13). Consider first the case |1+M|1. If μσ1, where σ1 is given by (4.6), then M0 and from (4.14) we obtain

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[1+2(1λ)t1+qt μ2(1λ)(1+q+q2t)([a]q)2 ([c]q)2(1+qt )2([a]q)2([c]q)2]

    which is the first part of the inequality (4.5). If μσ2, where σ2 is given by (4.7), then M2 and it follows from (4.14) that

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2[12(1λ)t1+qt +μ2(1λ)(1+q+q2t)([a]q)2 ([c]q)2(1+qt )2([a]q)2([c]q)2]

    and this is the third part of (4.5).

    Next, suppose σ1μσ2. Then, |1+M|1 and thus, from (4.14) we obtain

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2

    which is the second part of the inequality (4.5). In view of Lemma 2, the results are sharp for w(z)=z and w(z)=z2 or one of their rotations.

    For t=1 in Theorem 4, we state the following corollary.

    Corollary 6. Let fSaq(γ,λ) be given by (1.1) and let μ be a real number. Then

    |a3μa22|{2(1λ)cosγ([c]q)2q(1+q)([a]q)2[1+2(1λ)qμ2(1λ)(1+q)([a]q)2([c]q)2q([a]q)2([c]q)2](μσ3)2(1λ)cosγ([c]q)2q(1+q)([a]q)2(σ3μσ4)2(1λ)cosγ([c]q)2q(1+q)([a]q)2[12(1λ)q+μ2(1λ)(1+q)([a]q)2([c]q)2q([a]q)2([c]q)2](μσ4)

    where

    σ3=([a]q)2([c]q)2(1+q)([c]q)2([a]q)2,   σ4= (1+qλ)([a]q)2([c]q)2(1λ)(1+q)([a]q)2 ([c]q)2

    and all estimates are sharp.

    Taking t=0 in Theorem 4, we state the next corollary.

    Corollary 7. Let fKaq(γ,λ) be given by (1.1) and let μ be a real number. Then

    |a3μa22|{2(1λ)cosγ([c]q)2(1+q+q2)([a]q)2[1μ2(1λ)(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2](μ0)2(1λ)cosγ([c]q)2(1+q+q2)([a]q)2(0μσ5)2(1λ)cosγ([c]q)2(1+q+q2)([a]q)2[1+μ2(1λ)(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2](μσ5)

    where

    σ5= (1+q)2([a]q)2([c]q)2(1λ)(1+q+q2)([a]q)2 ([c]q)2

    and all estimates are sharp.

    We consider the Fekete-Szegö problem for the class Saq(γ,λ,t) with μ complex parameter.

    Theorem 5. Let fSaq(γ,λ,t) be given by (1.1) and let μ be a complex number. Then,

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2max{1,|2(1λ)cosγ 1+qt (μ(1+q+q2t)([a]q)2([c]q)2(1+qt)([a]q)2([c]q)2t)eiγ|}. (4.15)

    The result is sharp.

    Proof. Assume that fSaq(γ,λ,t). Making use of (4.10) and (4.11) we obtain

    |a3μa22|=2(1λ)cosγ([c]q)2(1+q+q2t)([a]q)2|w2[2eiγ(1λ)cosγ 1+qt (μ(1+q+q2t)([a]q)2([c]q)2(1+qt)([a]q)2([c]q)2t)1]w21|

    The inequality (4.15) follows as an application of Lemma 2 with

    s=2eiγ(1λ)cosγ 1+qt (μ(1+q+q2t)([a]q)2([c]q)2(1+qt)([a]q)2([c]q)2t)1.

    For t=1 in Theorem 5, we state the following corollary.

    Corollary 8. Let fSaq(γ,λ) be given by (1.1) and let μ be a complex number. Then,

    |a3μa22|2(1λ)cosγ([c]q)2q(1+q)([a]q)2max{1,|2(1λ)cosγ q (μ(1+q)([a]q)2([c]q)2([a]q)2([c]q)21)eiγ|}.

    The result is sharp.

    Taking t=0 in Theorem 5, we state the next corollary.

    Corollary 9. Let fKaq(γ,λ) be given by (1.1) and let μ be a complex number. Then,

    |a3μa22|2(1λ)cosγ([c]q)2(1+q+q2)([a]q)2max{1,|μ2(1λ)cosγ(1+q+q2)([a]q)2([c]q)2(1+q)2([a]q)2([c]q)2eiγ|}.

    The result is sharp.

    Utilizing the concepts of quantum calculus, we defined new subclass of analytic functions associated with q-analogue of Carlson-Shaffer operator. For this subclass we investigated some useful results such as coefficient estimates, subordination properties and Fekete-Szegö problem. Their are some problems open for researchers such as distortion theorems, closure theorems, convolution propertiies and radii problems. Moreover, these results can be extended to multivalent functions and meromophic functions.

    The authors are thankful to the referees for their valuable comments which helped in improving the paper.

    The authors declare that they have no competing interests.

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