The author constructs a new class SqλG,s,t of univalent functions applying the Ruscheweyh derivative. Moreover, the coefficient estimates including a Fekete-Szegö inequality of this class were determined.

Türkiye Bilimsel ve Teknolojik Arastirma KurumuTUBITAK 2214A1. Introduction and Definitions

Let Δ be the unit disk (1)z:z∈C,z<1,and let A be the class of functions analytic in Δ, satisfying the conditions(2)f0=0,f′0=1.Then each f∈A has the Taylor expansion(3)fz=z+∑k=2∞akzk.Moreover, by S we shall represent the class of all functions in A which are univalent in Δ. Let h(z) be an analytic function in Δ and h(z)≤1, such that(4)hz=h0+h1z+h2z2+⋯,where all coefficients are real. Also, let φ be an analytic and univalent function with positive real part in Δ with φ(0)=1, φ′(0)>0, and φ maps the unit disc Δ onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the form(5)φz=1+B1z+B2z2+⋯,where all coefficients are real and B1>0. Let P be the class of functions consisting of form (5).

If the functions f and g are analytic in Δ, then f is said to be subordinate to g, written as(6)fz≺gz,z∈Δif there exists a Schwarz function wz, analytic in Δ, with(7)w0=0,wz<1z∈Δsuch that (8)fz=gwzz∈Δ.

In the year 1970, Robertson [1] introduced the concept of quasi-subordination. For two analytic functions f and g, the function f is said to be quasi-subordinate to g in Δ and written as(9)fz≺qgzz∈Δif there exists an analytic function h(z)≤1 such that f(z)/h(z) analytic in Δ and(10)fzhz≺gzz∈Δ;that is, there exists a Schwarz function w(z) such that f(z)=h(z)g(w(z)). Observe that if h(z)=1, then f(z)=g(w(z)) so that f(z)≺g(z) in Δ. Also notice that if w(z)=z, then f(z)=h(z)g(z) and it is said that it is majorized by g and written by f(z)≪g(z) in Δ. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see, e.g., [1–3] for works related to quasi-subordination).

In [4], Sakaguchi introduced the class SS∗ of starlike functions with respect to symmetric points in Δ, consisting of functions f∈A that satisfy the condition Rzf′z/fz-f-z>0, z∈Δ. Similarly, in [5], Wang et al. introduced the class CS of convex functions with respect to symmetric points in Δ, consisting of functions f∈A that satisfy the condition Rzf′z′/f′z+f′-z>0,z∈Δ. For different parametric values, we get the classes studied in the literature by Frasin [6], Goyal et al. [7], and Owa et al. [8].

The Fekete-Szegö functional a3-μa22 for normalized univalent functions of the form given by (3) is well known for its rich history in Geometric Function Theory. Its origin was in the disproof by Fekete and Szegö [9] of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see, for details, [9]). The Fekete-Szegö functional has a3-μa22 since it received great attention, particularly in connection with many subclasses of the class of normalized analytic and univalent functions (see, e.g., [10–15]).

The object of the present paper is to introduce a new class of univalent functions applying the Ruscheweyh derivative, where Ruscheweyh [16] observed that(11)Dnfz=zzn-1fznn!for n∈N0=N∪0, where N=1,2,…. This symbol Dnf(z), n∈N0, is called by Al-Amiri [17] the nth order Ruscheweyh derivative of f(z). We note that D0f(z)=f(z), D1f(z)=zf′(z), and(12)Dnfz=z+∑k=2∞n+k-1nakzk.

2. Preliminary Results

The study of special functions plays an important role in Geometric Function Theory in Complex Analysis and its related fields. Special functions can be categorized into three, namely, Ramp function, threshold function, and sigmoid function. The popular type among all is the sigmoid function because of its gradient descendent learning algorithm. It can be evaluated in different ways, most especially by truncated series expansion. The sigmoid function of the form(13)κz=11+e-zis useful because it is differentiable. The Sigmoid function has very important properties, including the following (see [18]):

It outputs real numbers between 0 and 1.

It maps a very large input domain to a small range of outputs.

It never loses information because it is a one-to-one function.

It increases monotonically.

Lemma 1 (see [<xref ref-type="bibr" rid="B10">19</xref>]).

Let the Schwarz function w(z) be given by(14)wz=w1z+w1z2+⋯,z∈Δ; then (15)w1≤1,w2-ϑw12≤1+ϑ-1w12≤max1,ϑ,where ϑ∈C.

Lemma 2 (see [<xref ref-type="bibr" rid="B9">18</xref>]).

Let κ be a sigmoid function and (16)Gz=2κz=1+∑m=1∞-1m2m∑n=1∞-1nn!znm,and then G(z)∈P, z<1, where G(z) is a modified sigmoid function.

Lemma 3 (see [<xref ref-type="bibr" rid="B9">18</xref>]).

Let(17)Gn,mz=1+∑m=1∞-1m2m∑n=1∞-1nn!znm, and then Gn,m(z)<2.

3. Main Result and Its ConsequencesDefinition 4.

A function f∈A is said to be in the class SqλG,s,t, if the following quasi-subordination holds: (18)Dnfz′s-tzDnfsz-Dnftzλ-1≺qGz-1,z∈Δ,where s,t∈C with s≠t,t≤1 and λ≥0.

From the definition, it follows that f∈SqλG,s,t if and only if there exists an analytic function h(z) with h(z)≤1, such that(19)Dnfz′s-tz/Dnfsz-Dnftzλ-1hz≺Gz-1. If, in the subordination condition (19), h(z)=1, then the class SqλG,s,t is denoted by SλG,s,t and the functions therein satisfy the condition that(20)Dnfz′s-tzDnfsz-Dnftzλ≺Gz,z∈Δ.

Theorem 5.

Let f of the form (3) be in the class SqλG,s,t. Then (21)a2≤12n+12-λs+tand for some η∈C(22)a3-ηa22≤1n+1n+23-λs2+st+t2max1,14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+tand the result is sharp.

Proof.

Let f∈SqλG,s,t. In view of Definition 4, we can write(23)Dnfz′s-tzDnfsz-Dnftzλ-1≺hzGwz-1,where the function G(z) is a modified sigmoid function given by(24)Gz=1+12z-124z3+1240z5-164z6+77920160z7-⋯.Combining (4), (14), and (24), we obtain(25)hzGwz-1=12how1z+12how2+h1w1z2+⋯.In the light of (23) and (25), we get(26)n+12-λs+ta2=12how1,(27)n+1n+223-λs2+st+t2a3-λn+12s+t2-λ+1s+t2a22=12how2+12h1w1.Now, (26) gives(28)a2=how12n+12-λs+t. From (27), it follows that(29)a3=1n+1n+23-λs2+st+t2h1w1+how2+hoλ1+2-s+t/2-λs+ts+t42-λs+tw12.For some η∈C, we obtain from (28) and (29)(30)a3-ηa22=1n+1n+23-λs2+st+t2×how2+h1w1-14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+tho2w12.Since h(z) given by (4) is analytic and bounded in Δ, therefore, on using [20] (p 172), we have for some y (y≤1)(31)ho≤1,h1=1-ho2y. On putting the value of h1 from (31) into (30), we get(32)a3-ηa22=1n+1n+23-λs2+st+t2×how2+yw1-14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+tw12+yw1ho2.If ho=0 in (32), we obtain(33)a3-ηa22≤1n+1n+23-λs2+st+t2.If ho≠0 in (32), let(34)Tho=how2+yw1-14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+tw12+yw1ho2which is a polynomial in ho and hence analytic in ho≤1, and maximum T(ho) is attained at ho=eiθ, (0≤θ<2π). We find that(35)max0≤θ<2πTho=K1,a3-ηa22≤1n+1n+23-λs2+st+t2×w2-14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+tw12which on using Lemma 1 shows that(36)a3-ηa22≤1n+1n+23-λs2+st+t2max1,14ηn+23-λs2+st+t2n+12-λs+t2-λ1+2-s+t/2-λs+ts+t2-λs+t.

For the case when s=1, one has the following.

Corollary 6.

Let f of form (3) be in the class SqλG,1,t. Then(37)a2≤12n+12-λ1+tand for some η∈C(38)a3-ηa22≤1n+1n+23-λ1+t+t2max1,14ηn+23-λ1+t+t2n+12-λ1+t2-λ1+2-1+t/2-λ1+t1+t2-λ1+tand the result is sharp.

Putting t=-1 in Corollary 6, we obtain the following corollary.

Corollary 7.

Let f of form (3) be in the class SqλG,1,-1. Then (39)a2≤14n+1and for some η∈C(40)a3-ηa22≤1n+1n+23-λmax1,ηn+23-λ16n+1and the result is sharp.

Setting λ=0 in Corollary 7, we have the following.

Corollary 8.

Let f of form (3) be in the class SqG,1,-1. Then (41)a2≤14n+1and for some η∈C(42)a3-ηa22≤13n+1n+2max1,3ηn+216n+1and the result is sharp.

Setting λ=1 in Corollary 7, we have the following.

Corollary 9.

Let f of form (3) be in the class SqG,1,-1. Then (43)a2≤14n+1and for some η∈C(44)a3-ηa22≤12n+1n+2max1,ηn+28n+1and the result is sharp.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This research is supported by the Scientific and Technological Research Council of Turkey (TUBITAK 2214A).

RobertsonM. S.Quasi-subordination and coefficient conjecturesRenF. Y.OwaS.FukuiS.Some inequalities on quasi-subordinate functionsMacGregorT. H.Majorization by univalent functionsSakaguchiK.On a certain univalent mappingWangZ.-G.GaoC.-Y.YuanS.-M.On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric pointsFrasinB. A.Coefficient inequalities for certain classes of Sakaguchi type functionsGoyalS. P.VijaywargiyaP.GoswamiP.Sufficient conditions for Sakaguchi type functions of order βOwaS.SekineT.YamakawaR.On Sakaguchi type functionsFeketeM.SzegöG.Eine Bemerkung Uber UNGerade Schlichte FunktionenBucurR.AndreiL.BreazD.Coeffcient bounds and fekete-szegö problem for a class of analytic functions defined by using a new differential operatorBulutS.Fekete-Szegö problem for subclasses of analytic functions defined by Komatu integral operatorGoyalS. P.GoswamiP.Certain coefficient inequalities for Sakaguchi type functions and applications to fractional derivative operatorGoyalS. P.KumarR.Fekete-Szegö problem for a class of complex order related to Salagean operatorSakarF. M.AytaşS.GüneyH. Ö.On the Fekete-Szegö problem for generalized class Mα,γβ defined by differential operatorSrivastavaH. M.MishraA. K.DasM. K.The Fekete-Szegö problem for a subclass of close-to-convex functionsRuscheweyhS.New criteria for univalent functionsAl-AmiriH. S.On Ruscheweyh derivativesA Fadipe-JosephO.OladipoA. T.Uzoamaka EzeafulukweA.Modified Sigmoid function in univalent function theoryKeoghF. R.MerkesE. P.A coefficient inequality for certain classes of analytic functionsNehariZ.