Remarks on some starlike functions

R E S E A R C H

Open Access

Remarks on some starlike functions

Mamoru Nunokawa

_{and Janusz Sokół}

*_{Correspondence: [email protected]} 2_{Department of Mathematics,}

Rzeszów University of Technology, Al. Powsta ´nców Warszawy 12, Rzeszów, 35-959, Poland Full list of author information is available at the end of the article

Abstract

LetAbe the class of functions that are analytic in the unit diskD={z∈C:|z|< 1} and normalized byf(0) =f(0) – 1 = 0. In this work we investigate conditions under which|zf(z)/f(z) –

δ

δ

3z3+· · ·in the unit disc|z|< 1,

which satisfy|f(z) – 2|< 2. Furthermore, some geometric consequences of these results are given.

MSC: Primary 30C45; secondary 30C80

Keywords: differential subordinations; Nunokawa’s lemma; starlike functions of order alpha; convex functions of order alpha; strongly starlike functions of order alpha

1 Introduction

LetAbe the class of functions that are analytic in the unit diskD={z∈C:|z|< }and normalized byf() =f() –  = . The subclasses ofAconsisting of functions that are univalent inD, starlike with respect to the origin and convex will be denoted byS,S∗and

C, respectively. The classSα∗of starlike functions of orderα<  may be defined as

S∗

α=

f ∈A:Rezf _(z)

f(z) >α,z∈U

.

The classSα∗and the classCαof convex functions of orderα< 

Kα: =

f ∈A:Re

 +zf

_(z)

f(z)

>α,z∈U

=f ∈A:zf∈Sα∗

were introduced by Robertson in []. Ifα∈[; ), then a function in either of these sets is univalent. The convexity in one direction (it implies the univalence) of functions con-vex of negative order –/ was proved by Ozaki []. In [] Pfaltzgraffet al.established that the constant –/ is, in a certain sense, the best possible. A lot of the other equiva-lent/sufficient conditions for univalence or for the starlikeness, or more, for the convexity in one direction, one can find in []. In this work we consider a similar problem, namely findα,βsuch that

Re

 +zf

_(z)

f(z)

<α ⇒ zf

_(z)

f(z) –β

<β.

Ifβ∈(, ], it implies also the starlikeness off.

2 Preliminaries

The following lemma is a simple generalization of Nunokawa’s lemma [], which together with the lemma from [] has a surprising number of important applications in the theory of univalent functions.

Lemma . []Let p(z) =  + ∞_n=m≥cnzn_{be an analytic function inD.Suppose also that} there exists a point z∈Dsuch that

Rep(z)>  for|z|<|z|

and

Rep(z)=  and p(z)= .

Then we have

zp(z) p(z)

=ik,

where k is a real number and

k≥m



a+ a

≥m≥ whenArgp(z)

=π



and

k≤–m 

a+ a

≤–m≤– whenArgp(z)= –π ,

where|p(z)|=a.

3 Main results

Theorem . Assume thatδ≥/and m is a positive integer such that m> δ– .If f(z) =z+ ∞_n=manzn,and zf(z)/f(z)are analytic in the unit discDwith zf(z)= δf(z), f(z)= ,z∈Dand

Re

 +zf

_(z)

f(z)

<

⎧ ⎪ ⎨ ⎪ ⎩

δ+ (δ– /)(m– ) forδ∈[/, )and m≥δ/( –δ),m∈N, m–

(δ–) forδ≥and m> δ– ,m∈N, m–

(δ–) forδ∈[/, )andδ–  <m<δ/( –δ),m∈N, (.)

then we have

zf_f(z)(z)–δ<δ for|z|< .

Proof The functionzf(z)/f(z) is analytic inD, thus we can define the functionpby

zf(z) f(z) –δ=δ

p(z) +  – δ

p(z) –  + δ for|z|< , (.)

Then it follows that

 +zf

_(z)

f(z) =

δp(z) p(z) –  + δ +

δ–  p(z) –  + δ

zp(z)

p(z) . (.)

If there exists a pointz∈Dsuch that

zf_f(z)(z)–δ<δ for|z|<|z|

and

zf_f(z(z)₎ –δ=δ,

then by (.)

Rep(z)>  for|z|<|z|

and

Rep(z)= 

andp(z)=  by (.). Then applying Lemma ., we have

zp(z) p(z) =ik,

where

k≥(m– )(a _{+ )}

a whenArg

p(z)=π

 (.)

and

k≤–(m– )(a _{+ )}

a whenArg

p(z)

= –π ,

and wherep(z) =±iaand  <a. For the caseArg{p(z)}=π/, p(z) =iaand  <ait follows from (.) that

Re

 +zf

_(z)

f(z)

=Re δia

ia–  + δ +Re

(δ– )ik ia–  + δ

= a

_δ

a_{+ (δ– )} +

(δ– )ak a_{+ (δ– )}

=a

_{δ+ (δ– )ak}

Therefore, we have from (.)

Re

 +zf

_(z

) f(z)

≥aδ+ (δ– )a m–

 a_+

a a_{+ (δ– )}

=a

_{δ+ (δ– )(m– )(a}_{+ )}

(a_{+ (δ– )}₎

= δ+ (δ– /)(m– ) + δ(δ– )(m– )( –δ) – (δ– )

a_{+ (δ– )} fora> .

In the last expression, the numerator (m– )( –δ) – (δ– ) is nonnegative if and only if

δ∈[/, ) andm≥δ/( –δ) but this expression tends to +_{whena→ ∞. Therefore, in} this case we have

Re

 +zf

_(z

) f(z)

≥δ+ (δ– /)(m– ) forδ∈[/, ) andm>δ/( –δ). (.)

Furthermore, the numerator (m– )( –δ) – (δ– ) is negative if and only ifδ≥ and m∈Norδ∈[/, ) andm<δ/( –δ). In this case the quotient decreases whena→+_. Therefore, in this case we have

Re

 +zf

_(z)

f(z)

≥δ+ (δ– /)(m– ) +δ{(m– )( –δ) – (δ– )} δ– 

= m– 

(δ– ). (.)

We have assumed thatm> δ–  to have the right-hand side in (.) greater to . So in this case we have

δ–  <m< δ

 –δ forδ∈[/, ). (.)

Therefore, we can write (.) in the form

Re

 +zf

_(z)

f(z)

≥ m–  (δ– )

either for δ≥ andm> δ– ,m∈N,

or for δ∈[/, ) and δ–  <m<δ/( –δ). (.)

Inequalities (.) and (.) contradict the hypothesis of Theorem ., and therefore we have

Rep(z)>  for|z|< . (.)

Furthermore, from (.) and (.) we obtain

zf_f(z)(z)–δ=δp(z) +  – δ

p(z) –  + δ

For the case Arg{p(z)}= –π/, p(z) = –ia and  <a, applying the same method as above, we also have (.). Therefore, we get (.), which completes the proof of

Theo-rem ..

Substitutingδ=  in Theorem . leads to the following corollary.

Corollary . If f(z) =z+ ∞_n=manzn_{is analytic in the unit discDand}

Re

 +zf

_(z)

f(z)

<m– 

 ,

then we have

zf_f(z)(z)– <  for|z|< .

Substitutingδ= /,m=  in Theorem . gives the following corollary.

Corollary . If f(z) =z+ ∞_n=anzn_{is analytic in the unit discDand}

Re

 +zf

_(z)

f(z)

< ,

then we have

zf_f(z)(z)– 

<

 for|z|< .

Substitutingδ= /,m=  in Theorem . gives the following corollary.

Corollary . If f(z) =z+ ∞_n=anznis analytic in the unit discDand

Re

 +zf

_(z)

f(z)

< ,

then we have

zf_f(z)(z)– 

<

 for|z|< .

As a supplement to the above results recall here the known result [, p.] that iff(z) = z+ ∞_n=anznis analytic in the unit discDand

Re

 +zf

_(z)

f(z)

≺R,(z) = +z  –z+

z  –z,

then

Rezf _(z)

f(z) >  for|z|< .

Theorem . Let f(z) =z₊ ∞

n=anznbe analytic in the unit discD.If

f(z) – <  for|z|< , (.)

then

f(z)_z – <  for|z|< . (.)

Proof By the Schwarz lemma we have

fteiϕ– ≤t, t∈[, ).

Letz=reiϕ_{,r∈[, ), and letϕbe fixed. Using this we obtain}

f(z)_z – = |f

_{(z) – z|}

|z|

= |

z

(f(u) – ) du| |z|

= |

r (f(tei

ϕ_{) – ) d(te}iϕ_)|

|reiϕ|

= |

r e

iϕ_(f(teiϕ_{) – ) dt|}

|reiϕ|

≤

r |ei

ϕ_(f(teiϕ_{) – )|dt}

|reiϕ|

≤

r tdt

r

= r



r =r< .

Therefore, we obtain (.).

For the functionf(z) =z_{/ +z}_{, condition (.) is satisfied while (.) becomes|z|< } in the unit disc, which shows that the constant  in (.) cannot be replaced by a smaller one. A simple geometric observation yields the following corollary.

Corollary . Let f(z) =z₊ ∞

n=anznbe analytic in the unit discD.If

f(z) – <  for|z|< , (.)

then

Arg

f(z) z

<π

 for|z|< . (.)

Theorem . Let f(z) =z₊ ∞

n=anznbe analytic in the unit discD.If

f(z) – <  for|z|< , (.)

then

f(z)

z – 

<

 for|z|< . (.)

For the functionf(z) =z_{/ +z}_{, condition (.) is satisfied while (.) becomes|z/|<} / in the unit disc, which shows that the constant / in (.) cannot be replaced by a smaller one. A simple geometric observation yields the following corollary.

Corollary . Let f(z) =z+ ∞_n=anznbe analytic in the unit discD.If

f(z) – <  for|z|< , (.)

then

Arg

f(z) z

<sin–

 for|z|< . (.)

Using Corollaries . and . together, we obtain the next one.

Corollary . Let f(z) =z₊ ∞

n=anznbe analytic in the unit discD.If

f(z) – <  for|z|< , (.)

then

Arg

zf(z) f(z)

<π

 +sin –

≈. for|z|< . (.)

Proof From (.) and from (.), we have

Arg

zf(z) f(z)

=Arg

f(z) z

z f(z)

≤Arg

f(z) z

+Arg

f(z) z

< π  +sin

– 

≈..

Recall the classSS∗(β) of strongly starlike functions of orderβ,  <β≤,

SS∗_{(β) :=f∈A:Arg}zf(z)

f(z)

<βπ

 ,z∈U

,

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors jointly worked on the results, and they read and approved the final manuscript. Author details

1_{University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan.}2_{Department of Mathematics, Rzeszów}

University of Technology, Al. Powsta ´nców Warszawy 12, Rzeszów, 35-959, Poland. Received: 12 November 2013 Accepted: 5 December 2013 Published:30 Dec 2013 References

1. Robertson, MS: On the theory of univalent functions. Ann. Math.37, 374-408 (1936)

2. Ozaki, S: On the theory of multivalent functions. Sci. Rep. Tokyo Bunrika Daigaku4, 45-86 (1941)

3. Pfaltzgraff, JA, Reade, MO, Umezawa, T: Sufficient conditions for univalence. Ann. Fac. Sci. Kinshasa Zaïre Sect. Math.-Phys.2, 94-100 (1976)

4. Nunokawa, M: On properties of non-Carathéodory functions. Proc. Jpn. Acad. Ser. A68(6), 152-153 (1992) 5. Nunokawa, M: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad. Ser. A69(7),

234-237 (1993)

6. Nunokawa, M, Sokół, J: New conditions for starlikeness and strongly starlikeness of order alpha (submitted) 7. Miller, SS, Mocanu, PT: Differential Subordinations: Theory and Applications. Series of Monographs and Textbooks in

Pure and Applied Mathematics, vol. 225. Dekker, New York (2000)

8. Stankiewicz, J: Quelques problèmes extrêmaux dans les classes des fonctionsα-angulairement étoilées. Ann. Univ. Mariae Curie-Skłodowska, Sect. A20, 59-75 (1966)

9. Brannan, DA, Kirwan, WE: On some classes of bounded univalent functions. J. Lond. Math. Soc.1(2), 431-443 (1969) 10.1186/1029-242X-2013-593