Mathematical Problems in Engineering

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Internat. J. Math. & Math. Sci.

Vol. 24, No. 7 (2000) 433–435 S0161171200004634

A SUBORDINATION THEOREM FOR SPIRALLIKE FUNCTIONS

SUKHJIT SINGH (Received 24 November 1999)

Abstract.We prove a subordination relation for a subclass of the class ofλ-spirallike functions.

Keywords and phrases. Convex function, spirallike function, subordinating factor sequence.

2000 Mathematics Subject Classiﬁcation. Primary 30C45; Secondary 30C50.

1. Introduction. LetKdenote the usual class of convex functions. Denote bySp(λ),

−π/2< λ < π/2, the class of functionsf (z)=z+a2z²+···which are analytic inE and satisfy therein the condition

Re

e^iλzf(z) f (z)

>0. (1.1)

Spacek [3] proved that members ofSp(λ), known asλspirallike functions, are univalent inE. In 1989, Silverman [2] proved that if

∞ n=2

1+(n−1)secλan≤1

|λ|<π 2

, (1.2)

then the functionf (z)=z+ ^∞_n=2anzⁿbelongs toSp(λ). Let us denote byG(λ),the class of functionf (z)=z+ ^∞_n=2anzⁿwhose coeﬃcients satisfy the condition (1.2).

Note that G(0)is a subclass of the class of starlike functions (with respect to the origin) (see Silverman [1]).

In this paper, we prove a subordination theorem for the classG(λ). To state and prove our main result we need the following deﬁnitions and lemma.

Deﬁnition1.1. Iff (z)= ^∞_n=0anzⁿandg(z)= ^∞_n=0bnzⁿare analytic in|z|< r, then their Hadamard product/convolution,f∗gis the function deﬁned by the power series

(f∗g)(z)= ∞ n=0

anbnzⁿ. (1.3)

The functionf∗gis also analytic in|z|< r.

Deﬁnition1.2. Letf be analytic inE, ganalytic and univalent inE andf (0)= g(0). Then by the symbolf (z)≺g(z)(fsubordinate tog) inE, we shall mean that f (E)⊂g(E).

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434 SUKHJIT SINGH

Deﬁnition1.3. A sequence{bn}^∞1 of complex numbers is said to be a subordinating factor sequence if wheneverf (z)= ^∞_k=1akz^k, a1=1 is regular, univalent and convex inE, we have

∞ k=1

bkakz^k≺f (z) inE. (1.4)

Lemma1.4. The sequence{bn}^∞₁ is a subordinating factor sequence if and only if Re

1+2

∞ n=1

bnzⁿ

>0, (z∈E). (1.5)

This lemma which gives a beautiful characterisation of a subordinating factor sequence is due to Wilf [4].

2. Main theorem

Theorem2.1. Letf∈G(λ). Then 1+secλ

2(2+secλ)(f∗g)(z)≺g(z), (z∈E) (2.1) for every functiongin the classK.

In particular

Ref (z) >− 2+secλ

(1+secλ), (z∈E). (2.2)

The constant(1+secλ)/2(2+secλ)cannot be replaced by any larger one.

Takingλ=0, we obtain the following corollary.

Corollary2.2. Iff (z)=z+a2z²+ ··· is regular inE and satisﬁes therein the condition

∞ n=2

nan≤1, (2.3)

then for every functionginK, we have 1

3(f∗g)(z)≺g(z), (|z|<1). (2.4) In particular,Ref (z) >−3/2, z∈E. The constant1/3is best possible.

Proof of Theorem2.1. Let f (z)=z+ ^∞_n=2anzⁿ be any member of the class G(λ)and letg(z)=z+ ^∞_n=2cnzⁿbe any function in the classK. Then

1+secλ

2(2+secλ)(f∗g)(z)= 1+secλ 2(2+secλ)

z+

∞ n=2

ancnzⁿ

. (2.5)

Thus, by Deﬁnition 1.3, the assertion of our theorem will hold if the sequence (1+secλ)an

2(2+secλ) ∞

n=1

(2.6) is a subordinating factor sequence, witha1=1. In view of the lemma, this will be the

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A SUBORDINATION THEOREM FOR SPIRALLIKE FUNCTIONS 435 case if and only if

Re

1+2 ∞ n=1

1+secλ 2(2+secλ)anzⁿ

>0, (z∈E). (2.7) Now

Re

1+1+secλ 2+secλ

∞ n=1

anzⁿ

=Re

1+1+secλ

2+secλz+ 1 2+secλ

∞ n=2

(1+secλ)anzⁿ

>

1−1+secλ

2+secλr− 1 2+secλ

∞ n=2

1+(n−1)secλanrⁿ

(because 1+secλ≤1+(n−1)secλfor alln≥2,|λ|< π/2)

>

1−1+secλ

2+secλr− 1 2+secλr

(|z| =r )

>0.

(2.8)

Thus (2.7) holds true in E. This proves the ﬁrst assertion. That Ref (z) > −(2+ secλ)/(1+secλ)forf∈G(λ)follows by takingg(z)=z/(1−z)in (2.1). To prove the sharpness of the constant(1+secλ)/2(2+secλ), we consider the functionf0deﬁned byf0(z)=z−(1/(1+secλ))z²(|λ|< π/2), which is a member of the classG(λ). Thus from the relation (2.1) we obtain

1+secλ

2(2+secλ)f0(z)≺ z

1−z. (2.9)

It can be easily veriﬁed that

|z|≤1minRe

1+secλ 2(2+secλ)f0(z)

= −1

2. (2.10)

This shows that the constant(1+secλ)/2(2+secλ)is best possible.

Acknowledgement. This paper was presented at the 62nd annual conference of the Indian Mathematical Society held at IIT, Kanpur from December 22–25, 1996.

The author is thankful to Prof. Ram Singh, Department of Mathematics, Punjabi University, Patiala, for his help and encouragement during this work.

References

[1] H. Silverman,Univalent functions with negative coeﬃcients, Proc. Amer. Math. Soc.51 (1975), 109–116. MR 51#5910. Zbl 311.30007.

[2] , Suﬃcient conditions for spiral-likeness, Int. J. Math. Math. Sci.12(1989), no. 4, 641–644. MR 90k:30024. Zbl 688.30009.

[3] L. Spacek,Contribution à la theorie des fonctions univalents, Cas. Mat. Fys.62(1932), no. 2, 12–19 (Czech). Zbl 006.06403.

[4] H. S. Wilf,Subordinating factor sequences for convex maps of the unit circle, Proc. Amer.

Math. Soc.12(1961), 689–693. MR 23#A2519. Zbl 100.07201.

Sukhjit Singh: Department of Mathematics, Sant Longowal Institute of Engineering

& Technology, Longowal-148 106(Punjab), India

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