On some geometric properties of multivalent functions

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R E S E A R C H

Open Access

On some geometric properties of

multivalent functions

Mamoru Nunokawa

1

_{and Janusz Sokół}

2*

*_{Correspondence: jsokol@prz.edu.pl} 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

We prove some new suﬃcient conditions for a function to bep-valent orp-valently starlike in the unit disc.

MSC: Primary 30C45; secondary 30C80

Keywords: analytic functions;p-valent functions;p-valently starlike

1 Introduction

LetApbe the class of analytic functions of the form

f(z) =zp+

∞

n=

ap+nzp+n (z∈D). (.)

A functionf(z) which is analytic in a domainD⊂Cis calledp-valent inDif for every complex numberw, the equationf(z) =whas at mostproots inD, and there will be a complex numberwsuch that the equationf(z) =whas exactlyproots inD. Further,

a functionf ∈Ap,p∈N\ {}, is said to bep-valently starlike of orderα, ≤α<p, if

Re

zf(z)

f(z)

>α (z∈D).

The class of all such functions is usually denoted byS_p∗(α). Forp= , we receive the well-known class of normalized starlike univalent functionsS∗(α) of orderα,S_p∗() =S_p∗. If

zf(z)∈S∗(α), thenf(z) is said to bep-valently convex of orderα, ≤α<p. The class of all such functions is usually denoted byCp(α). Forp= , we receive the well-known class

of normalized convex univalent functionsC(α) of orderα,Cp() =Cp.

The well-known Noshiro-Warschawski univalence condition (see [] and []) indicates that iff(z) is analytic in a convex domainD⊂Cand

Reeiθf(z)>  (z∈D),

whereθis a real number, thenf(z) is univalent inD. In [] Ozaki extended the above result by showing that iff(z) of the form (.) is analytic in a convex domainDand for some real

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θ we have

Reeiθf(p)(z)> , |z|< ,

thenf(z) is at mostp-valent inD. Applying Ozaki’s theorem, we ﬁnd that iff(z)∈Apand

Ref(p)(z)>  (z∈D),

thenf(z) is at mostp-valent in|z|< . In [] it was proved that iff(z)∈Ap,p≥, and

argf(p)(z)<π

 , |z|< ,

thenf(z) is at mostp-valent in|z|< .

2 Preliminary lemmata

Lemma .[] Let f(z) =z+az+· · ·be analytic in the unit disc and suppose that

f(z)< , |z|< , (.)

then f(z)is univalent in|z|< .

Lemma .[] Let p(z)be an analytic function in|z|< with p() = ,p(z)= .If there exists a point z,|z|< ,such that

argp(z)<π α

 for|z|<|z|

and

argp(z)= π α



for some <α< ,then we have

zp(z) p(z)

=isα,

where

s≥



a+

a whenarg

p(z)

=π α 

and

s≤– 

a+

a whenarg

p(z)

= –π α  ,

where

p(z)

/α

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Lemma .[] Let p be a positive integer.If f(z) =zp₊∞

n=p+anznis analytic inDand if

it satisﬁes

Re

zf(p)(z)

f(p–)₍_z₎

> , z∈D, (.)

then f(z)is p-valently starlike inDand

Re

zf(k)(z)

f(k–)₍_z₎

> , z∈D (.)

for k= , , . . . , (p– ).

Lemma .([], p.) Let f ∈Ap.If there exists a(p–k+ )-valent starlike function

g(z) =∞_n₌_p_–_k_+bnzn(bp–k+= )that satisﬁes

Re

zf(k)₍_z₎

g(z)

> , |z|< , (.)

then f(z)is p-valent in|z|<  .

3 Main results

Now we state and prove the main results. The ﬁrst theorem poses a growth condition to a higher derivative of an analytic function. Hence, its proof, among others, uses the method of integrating the derivatives, which is frequently used in complex analysis, especially in estimating point evaluation operators (see,e.g., Lemma  in [] and Lemma  in []).

Theorem . Let f ∈Apand suppose that

f(p+)(z)<α(β)(p!), |z|< , (.)

whereβ= .· · ·is the positive root of the equation

β+ 

πtan

–_β_{= ,} _{ <}_β_{< } _(.)

and

α(β) =sin

π



β+ (/π)tan–β

=sin

( –β)π



= .· · ·.

Then we have

Re

zf(p)₍_z₎ f(p–)₍_z₎

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and,therefore,we have

Re

zf(z)

f(z)

> , |z|< , (.)

or f(z)is p-valently starlike in|z|< .

Proof From the hypothesis (.), we have

f(p)(z) –p!=

z



f(p+)(t) dt

≤ |z|



f(p+)(t)|dt|<p!

|z| 

α(β)|dt|

=p!α(β)|z|<p!α(β).

This shows that

argf(p)(z)<sin–α(β), |z|< . (.)

Let us put

q(z) =f

(p–)₍_z₎

p!z , q() = . (.)

Then it follows that

q(z) +zq(z) =f

(p)₍_z₎ p! .

If there exists a pointz,|z|< , such that

argq(z)<πβ

 for|z|<|z|

and

argq(z)= πβ

 ,

then by Lemma . we have

zq(z) q(z)

=iβk,

where

k≥ whenargq(z)

=πβ

 (.)

and

k≤– whenargq(z)

= –πβ

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For the case (.), we have

argf(p)(z)=arg

f(p)₍_z₎ p!

=argq(z) +zq(z)

=arg

q(z)

 +zq

₍_z ) q(z)

= πβ

 +arg{ +iβk}

≥πβ

 +tan

–_β 

= π 

β+ (/π)tan–(β)

=sin–α(β).

This contradicts (.), and for the case (.), applying the same method as above, we have

argf(p)(z)≤–sin–α(β).

This also contradicts (.) and, therefore, it shows that

arg

f(p–)₍_z₎

p!z

<πβ

 , |z|< . (.)

Applying (.) and (.), we have

arg

zf(p)(z)

f(p–)₍_z₎

=argf(p)(z)+arg

z f(p–)₍_z₎

≤argf(p)(z)+arg

z f(p–)₍_z₎

<sin–α(β) + πβ



= π 

β+



πtan –_β



= π .

This shows that

Re

zf(p)₍_z₎ f(p–)₍_z₎

> , |z|< ,

and by Lemma . we have

Re

zf(z)

f(z)

> , |z|< ,

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Remark Note that ifm(x) = x+_πtan–_x_{, then}

m(.) = .· · ·, m(.) = .· · ·.

Hence, if

β+



πtan –_β

= ,

thenβ= .· · ·. Moreover,

.· · ·<sin–α(β) = π



β+



πtan –_β

 =

( –β)π

 < .· · ·

and

α(β) =sin

π



β+

πtan

–_β _{= .}_{· · ·}_.

Forp= , Theorem . becomes the following corollary which extends the result contained in Lemma ..

Corollary . Let f ∈A()and suppose that

f(z)<α(β), |z|< , (.)

whereβ= .· · ·is the positive root of the equation

β+ 

πtan

–_β_{= ,} _{ <}_β_{< ,} _(.)

and

α(β) =sin

( –β)π



= .· · ·.

Then we have

Re

zf(z)

f(z)

> , |z|< , (.)

or f(z)is starlike univalent in|z|< .

An analytic functionf(z) is said to be typically real if the inequalityImzImf(z)≥ holds for allz∈D. From the deﬁnition of a typically real function it follows thatz∈D+_⇔_f₍_z₎_∈ C+_and_z_∈_D–_⇔_f₍_z₎_∈_C–_{. The symbols}_D+_,_D–_,_C+_,_C–_{stand for the following sets: the}

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Theorem . Let f(z)∈Apand suppose that

arg

zf(p)₍_z₎

g(z)

<π

 +tan

– –|z|

 +|z|, z∈D, (.)

where g(z)is univalent starlike inDand the functions

zg(z)

g(z) and

z(zf(p_g–)₍_z₎(z))

zf(p–)₍_z₎

g(z)

(.)

are typically real inD.Then we have

argzf (p–)₍_z₎

g(z)

<π

, z∈D. (.)

Proof Let us put

q(z) =zf

(p–)₍_z₎

p!g(z) , q() = .

Then it follows that

zf(p)₍_z₎

g(z) =p!q(z)

zg(z)

g(z) +

zq(z)

q(z) . (.)

From the hypothesis

zg(z)

g(z) +

zq(z)

q(z)

is typically real inD. If there exists a pointz,|z|< , such that

argq(z)<π

 for|z|<|z|

and

argq(z)= π

, q(z) =±ia, anda> ,

then by Lemma . we have

zq(z) q(z)

=is, (.)

where

s≥



a+

a whenarg

q(z)

=π 

and

s≤– 

a+

a whenarg

q(z)

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For the case

argq(z)

=π

, q(z) =ia,a> ,

we have

zq(z) q(z)

=is, s≥ 



a+

a > , (.)

hence

Im{z}>  (.)

becausezq(z)/q(z) is typically real. Therefore, (.) yields that

Im

zg(z)

g(z)

> , (.)

becausezg(z)/g(z) is typically real. Moreover,

 –|z|

 +|z|≤

Re

zg(z)

g(z)

≤ +|z|

 –|z|

(.)

becauseg(z) is a univalent starlike function, see []. Applying (.), (.) and (.) in (.), we have

arg

zf(p)(z) g(z)

=argq(z)

+arg

zg(z) g(z)

+zq

₍_z ) q(z)

=argq(z)

+arg

zg(z) g(z)

+is

≥ π

 +tan

– –|z|

 +|z|

.

This contradicts (.), and for the case

argq(z)

= –π ,

applying the same method as above, we have

arg

zf(p)(z) g(z)

=argq(z)

+arg

zg(z) g(z)

+zq

₍_z ) q(z)

≤–

π

 +tan

– –|z|

 +|z|

.

This also contradicts (.) and, therefore, it shows that (.) holds.

Corollary . Let f(z)∈Apand all the coeﬃcients of f(z)are real and suppose that

arg

zf(p)₍_z₎

g(z)

<π

 +tan

– –|z|

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where g(z)is univalent starlike and typically real inD.Then we have

argzf

(p–)₍_z₎ g(z)

<π

, z∈D.

Competing interests

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

All authors contributed equally to the writing of this paper. All authors 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: 26 June 2015 Accepted: 10 September 2015

References

1. Noshiro, K: On the theory of Schlicht functions. J. Fac. Sci., Hokkaido Univ., Ser. 12(3), 129-135 (1934)

2. Warschawski, S: On the higher derivatives at the boundary in conformal mapping. Trans. Am. Math. Soc.38, 310-340 (1935)

3. Ozaki, S: On the theory of multivalent functions. Sci. Rep. Tokyo Bunrika Daigaku, Sect. A2, 167-188 (1935) 4. Nunokawa, M: A note on multivalent functions. Tsukuba J. Math.13(2), 453-455 (1989)

5. Ozaki, S, Ono, I, Umezawa, T: On a general second order derivative. Sci. Rep. Tokyo Bunrika Daigaku, Sect. A124(5), 111-114 (1956)

6. Nunokawa, M: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 69(7), 234-237 (1993)

7. Nunokawa, M: On the theory of multivalent functions. Tsukuba J. Math.11(2), 273-286 (1987)

8. Krantz, S, Stevi´c, S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. TMA71, 1772-1795 (2009)

9. Stevi´c, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. TMA71, 6323-6342 (2009)