Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 DOI 10.1186/s13660-016-1034-z

R E S E A R C H Open Access

Conditions for one direction convexityand starlikenessMamoru Nunokawa1, Oh Sang Kwon2, Young Jae Sim2, Ji Hyang Park3 and Nak Eun Cho3*

*Correspondence:necho@pknu.ac.kr3Department of AppliedMathematics, Pukyong NationalUniversity, Busan, 48513, KoreaFull list of author information isavailable at the end of the article

AbstractWe investigate several sufficient conditions on a function to be convex in onedirection or starlike in one direction.

MSC: Primary 30C45

Keywords: analytic functions; univalent functions; convex functions in onedirection; starlike functions in one direction

1 IntroductionLetH denote the class of functions analytic in the unit diskD := {z ∈ C : |z| < }, and denoteby A the class of analytic functions in H that are normalized by f () = = f ′() – . Also,let S denote the subclass of A composed of functions that are univalent in D.

We say that a function f is starlike in one direction if f it maps |z| = r for every r near onto a contour C that is cut by a straight-line passing through the origin in two and nomore than two points. Robertson [] found the following sufficient condition for starlike-ness in one direction.

Lemma Let f (z) be analytic in |z| ≤ r, and f (z) �= in < |z| ≤ r. Further, let f () = .Suppose that

∫ π

∣∣∣∣Rzf′(z)

f (z)

∣∣∣∣dθ < π , z = ρeiθ , for every ρ ≤ r.

Then, for every ρ ≤ r, f (z) maps |z| = ρ onto a curve that is starlike in one direction.

A function is said to be convex in one direction in |z| < r (r > ) if the function maps|z| = ρ < r for every ρ near r into a contour that may be cut by every straight-line parallelto this direction in no more than two points. It is known (see []) that if f ∈A and zf ′(z) isstarlike in one direction, then f (z) is convex in one direction and belongs to S . Therefore,we can obtain the following lemma (see also [–]).

Lemma Let f (z) = z+∑∞

n= anzn be analytic for |z| ≤ and f ′(z) �= on |z| = r < . Supposethat

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Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 Page 2 of 6

∫ π

∣∣∣∣ + R{

zf ′′(z)f ′(z)

}∣∣∣∣dθ < π , z = reiθ , for every r < .

Then f (z) is convex in one direction, and hence f (z) is univalent in |z| ≤ .

We may refer to [–] for more sufficient conditions on analytic functions to be convexin one direction.

In the present paper, we investigate several sufficient conditions on functions in A tobe convex in one direction using various methods. Also, we find sufficient conditions forstarlikeness in one direction.

2 Main resultsTheorem Let f (z) ∈A and suppose that

∣∣∣∣ + R{

zf ′′(z)f ′(z)

}∣∣∣∣ < R{

+ z – z

}(z ∈D). ()

Then f (z) is convex in one direction, and hence f (z) is univalent in D.

Proof Let ≤ r < . From hypothesis () we have∫ π

∣∣∣∣ + R{

zf ′′(z)f ′(z)

}∣∣∣∣dθ

< ∫

|z|=r

{R

{ + z – z

}}dθ

= ∫ π

– r

– r cos θ + rdθ

= π .

Therefore, by Lemma , f (z) is convex in one direction in D. �

Example Consider the function f : D →C defined by f(z) = z/( – z). Then we have

+zf ′′ (z)f ′ (z)

= + z – z

(z ∈D).

Moreocer, we can easily check that condition () holds for the function f. Therefore, byTheorem the function f is convex in one direction and univalent in D.

Theorem Let f (z) ∈A and suppose that∣∣∣∣ + zf

′′(z)f ′(z)

∣∣∣∣ ≤∣∣∣∣ + z – z

∣∣∣∣ (z ∈D). ()

Then f (z) is convex in one direction in |z| < r = . · · · , where r is the root of theequation

π(

+ r – r

)+ log

( + r – r

)= π . ()

Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 Page 3 of 6

Proof Let ≤ r < . From inequality () we have∫ π

∣∣∣∣ + R{

zf ′′(z)f ′(z)

}∣∣∣∣dθ

≤∫ π

∣∣∣∣ + zf′′(z)

f ′(z)

∣∣∣∣dθ

≤∫ π

∣∣∣∣ + z – z∣∣∣∣dθ

≤∫ π

(∣∣∣∣ – r

– r cos θ + r

∣∣∣∣ +∣∣∣∣ r sin θ – r cos θ + r

∣∣∣∣)

dθ

≤ π(

+ r – r

)+ log

( + r – r

).

Define the function g : [, ) →R by

g(r) = π(

+ r – r

)+ log

( + r – r

).

Then g() = π and g(r) → ∞ as r → –. Also, we have that the function g is increasingon [, ) since

g ′(r) =π

( – r)+

– r

>

for all r ∈ [, ). Therefore, there exists a unique root r in [, ) such that g(r) = π . Hence,we have

∫ π

∣∣∣∣ + R{

zf ′′(z)f ′(z)

}∣∣∣∣dθ < π

for |z| < r. It follows from Lemma that f (z) is convex in one direction in |z| < r. �

Theorem Let f (z) ∈A and suppose that∣∣∣∣R

{ +

zf ′′(z)f ′(z)

}–

∣∣∣∣ < R{

+ z – z

}+

(z ∈D). ()

Then f (z) is convex in one direction, and hence f (z) is univalent in D.

Proof Let ≤ r < . From hypothesis () we have∫

|z|=r

{∣∣∣∣R{

+zf ′′(z)f ′(z)

}∣∣∣∣ – }

dθ

≤∫

|z|=r

∣∣∣∣R{

+zf ′′(z)f ′(z)

}–

∣∣∣∣dθ

<∫

|z|=r

{R

{ + z – z

}+

}dθ

= π .

Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 Page 4 of 6

Therefore, we have∫

|z|=r

∣∣∣∣R{

+zf ′′(z)f ′(z)

}∣∣∣∣dθ < π

for |z| = r < . This shows that f (z) is convex in one direction in D. �

Corollary Let f (z) ∈A and suppose that∣∣∣∣R

{ +

zf ′′(z)f ′(z)

}∣∣∣∣ < R{

+ z – z

}+ (z ∈ D).

Then f (z) is convex in one direction, and hence f (z) is univalent in D.

Theorem Let f (z) ∈A and suppose that∣∣∣∣ + zf

′′(z)f ′(z)

∣∣∣∣ <√

(z ∈D). ()

Then f (z) is convex in one direction, and hence f (z) is univalent in D.

Proof Let ≤ r < . From () we have∫

|z|=r

∣∣∣∣ + zf′′(z)

f ′(z)

∣∣∣∣

dθ < π . ()

Note that

∫|z|=r

( +

zf ′′(z)f ′(z)

)dθ =

∫|z|=r

{ +

zf ′′(z)f ′(z)

+(

zf ′′(z)f ′(z)

)}dθ = π ()

and

∫|z|=r

( +

zf ′′(z)f ′(z)

)dθ = π . ()

Therefore, from (), (), and () we have

∫|z|=r

( + R

zf ′′(z)f ′(z)

)dθ

=

∫|z|=r

[( +

zf ′′(z)f ′(z)

)+

∣∣∣∣ + zf′′(z)

f ′(z)

∣∣∣∣

+(

+zf ′′(z)f ′(z)

)]dθ

< π . ()

Hence, applying the Cauchy-Schwarz inequality and (), we get

∫|z|=r

∣∣∣∣ + Rzf′′(z)

f ′(z)

∣∣∣∣dθ ≤√

π∫

|z|=r

∣∣∣∣ + Rzf′′(z)

f ′(z)

∣∣∣∣

dθ < π .

This completes the proof of Theorem . �

Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 Page 5 of 6

Example Consider the function f : D →C defined by

f(z) =√

((z +

√√

)–

)(z ∈D).

Then we have

∣∣∣∣ + zf′′

(z)f ′(z)

∣∣∣∣ =∣∣∣∣√

√

z + √ + z

∣∣∣∣ <√

(z ∈D).

Hence, it follows from Theorem that the function f is convex in one direction. In fact,the function f is convex in the direction of the positive real axis.

Applying the same method as that used in the proof of the aforementioned theorems andLemma , we have the following sufficient conditions on analytic functions to be starlikein one direction.

Theorem Let f (z) ∈A and suppose that∣∣∣∣R

{zf ′(z)f (z)

}∣∣∣∣ < R{

+ z – z

}(z ∈D).

Then f (z) is starlike in one direction in D.

Theorem Let f (z) ∈A and suppose that∣∣∣∣zf

′(z)f (z)

∣∣∣∣ ≤∣∣∣∣ + z – z

∣∣∣∣ (z ∈D).

Then f (z) is starlike in one direction in |z| < r = . · · · , where r is the root of equa-tion ().

Theorem Let f (z) ∈A and suppose that∣∣∣∣R

{zf ′(z)f (z)

}–

∣∣∣∣ < R{

+ z – z

}+

(z ∈D).

Then f (z) is starlike in one direction in D.

Theorem Let f (z) ∈A and suppose that∣∣∣∣zf

′(z)f (z)

∣∣∣∣ <√

(z ∈D).

Then f (z) is starlike in one direction in D.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors jointly worked on the results and they read and approved the final manuscript.

Nunokawa et al. Journal of Inequalities and Applications (2016) 2016:90 Page 6 of 6

Author details1University of Gunma, Hoshikuki-Cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan. 2Department of Mathematics,Kyungsung University, Busan, 48434, Korea. 3Department of Applied Mathematics, Pukyong National University, Busan,48513, Korea.

AcknowledgementsThe authors would like to express their thanks to the editor Professor S Stevic and the referees for many valuable advicesregarding a previous version of this paper. This research was supported by the Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(No. 2011-0007037).

Received: 26 January 2016 Accepted: 2 March 2016

References1. Robertson, MS: Analytic functions star-like in one direction. Am. J. Math. 58, 465-472 (1936)2. Ozaki, S: On the theory of multivalent functions, II. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 4, 45-86 (1941)3. Umezawa, T: On the theory of univalent functions. Tohoku Math. J. 7, 212-228 (1955)4. Umezawa, T: Analytic functions convex in one direction. J. Math. Soc. Jpn. 4(2), 194-202 (1952)5. Nunokawa, M, Sokół, J: Some simple conditions for univalence. C. R. Math. Acad. Sci. Soc. R. Can. 354, 7-11 (2015).

doi:10.1016/j.crma.2015.10.0016. Ogawa, S: Some criteria for univalence. J. Nara Gakugei Univ. 1(10), 7-12 (1961)7. Shah, GM: On holomorphic functions convex in one direction. J. Indian Math. Soc. 37, 257-276 (1973)

http://dx.doi.org/10.1016/j.crma.2015.10.001

Conditions for one direction convexity and starlikenessAbstractMSCKeywords

IntroductionMain resultsCompeting interestsAuthors' contributionsAuthor detailsAcknowledgementsReferences