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Article

# The Order of Strongly Starlikeness of the Generalized α-Convex Functions

by
Yuan Yuan
1,
Rekha Srivastava
2,* and
Jin-Lin Liu
3,*
1
Department of Mathematics, Maanshan Teacher’s College, Maanshan 243000, China
2
Department of Mathematics and Statistics, University of Victoria, British Columbia, VIC V8W 3R4, Canada
3
Department of Mathematics, Yangzhou University, Yangzhou 225002, China
*
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(1), 76; https://doi.org/10.3390/sym11010076
Submission received: 28 December 2018 / Revised: 8 January 2019 / Accepted: 10 January 2019 / Published: 11 January 2019
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

## Abstract

:
We consider the order of the strongly-starlikeness of the generalized $α$-convex functions. Some sufficient conditions for functions to be p-valently strongly-starlike are given.
MSC:
30C45; 30C80

## 1. Introduction

Let $N$, $R$ and $C$ denote the sets of positive integers, real numbers and complex numbers, respectively.
Definition 1.
A function f is called p-valent in a domain $D ⊂ C$ if the equation $f ( z ) = w$ has at most p roots in $D$ for every complex number w, and there is a complex number $w 0$ such that $f ( z ) = w 0$ has exactly p roots in $D$.
Let $A ( p )$ denote the class of analytic functions in $U = { z : z ∈ C a n d | z | < 1 }$ of the form:
$f ( z ) = z p + ∑ n = 1 ∞ a p + n z p + n ( p ∈ N ) .$
For $p = 1$, we denote $A : = A ( 1 )$.
Definition 2.
A function $f ∈ A ( p )$ is said to be p-valently starlike in $U$ if it satisfies:
$Re z f ′ ( z ) f ( z ) > 0 ( z ∈ U ) .$
We denote by $S p *$ the subclass of $A ( p )$ consisting of all p-valently starlike functions in $U$.
Definition 3.
If $f ∈ A ( p )$ satisfies:
$arg z f ′ ( z ) f ( z ) < β π 2 ( z ∈ U )$
for some $β ∈ ( 0 , 1 ]$, then the function f is called p-valently strongly-starlike of order β in $U$. We denote this class by $SS p * ( β )$.
For $p = 1$, the class $SS 1 * ( β )$ was introduced by Brannan and Kirwan [1]. It is clear that:
$SS p * ( β ) ⊂ S p * and SS p * ( 1 ) = S p * .$
The strongly-starlike functions and related functions have been extensively studied by several authors (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]).
We say that for functions f and g analytic in $U$, g is subordinate to f, written $g ≺ f$, if there exists a Schwarz function w such that $g ( z ) = f ( w ( z ) )$ for $z ∈ U$. In particular, if f is univalent in $U$, then:
$g ( z ) ≺ f ( z ) ( z ∈ U ) ⟺ g ( 0 ) = f ( 0 ) and g ( U ) ⊂ f ( U ) .$
In [17], Mocanu first introduced the class:
$M ( α ) = f ∈ A : Re α 1 + z f ″ ( z ) f ′ ( z ) + ( 1 − α ) z f ′ ( z ) f ( z ) > 0 , α ∈ R , z ∈ U$
of $α$-convex functions, which give a continuous passage from convex to starlike functions. He proved that every $α$-convex function is starlike. Recently, Nunokawa, Sokól and Trabka-Wieclaw [8] considered the generalized $α$-convex function class:
$M ( α , β ) = f ∈ A : arg α 1 + z f ″ ( z ) f ′ ( z ) + ( 1 − α ) z f ′ ( z ) f ( z ) < β π 2 , α ∈ R , β ∈ ( 0 , 1 ] , z ∈ U .$
In this paper, we shall further study the properties of the generalized $α$-convex functions. Several sufficient conditions for functions to be p-valently strongly starlike are obtained.
The following lemmas will be required in our investigation.
Lemma 1 (See [18])
Let g be analytic and univalent in $U$. Furthermore, let θ and φ be analytic in a domain $D ⊇ g ( U )$ with $φ ( w ) ≠ 0$ for $w ∈ g ( U )$. Put:
$Q ( z ) = z g ′ ( z ) φ ( g ( z ) ) and h ( z ) = θ ( g ( z ) ) + Q ( z )$
and suppose that
(i) Q is univalent starlike in $U$ and
(ii) $Re z h ′ ( z ) Q ( z ) = Re θ ′ ( g ( z ) ) φ ( g ( z ) ) + z Q ′ ( z ) Q ( z ) > 0 ( z ∈ U ) .$
If q is analytic in $U$ with $q ( 0 ) = g ( 0 )$, $q ( U ) ⊂ D$ and:
$θ ( q ( z ) ) + z q ′ ( z ) φ ( q ( z ) ) ≺ θ ( g ( z ) ) + z g ′ ( z ) φ ( g ( z ) ) = h ( z ) ( z ∈ U ) ,$
then $q ( z ) ≺ g ( z )$$( z ∈ U )$. The function g is the best dominant of (5).
Lemma 2 (See [19])
Let $p ( z )$ be an analytic function in $U$ of the form:
$p ( z ) = 1 + ∑ n = m ∞ c n z n , c m ≠ 0 , m ≥ 1 ,$
with $p ( z ) ≠ 0$ in $U$. If there exists a point $z 0$, $| z 0 | < 1$, such that:
$arg { p ( z ) } < π 2$
for $| z | < | z 0 |$ and:
$arg { p ( z 0 ) } = π 2 ,$
then:
$z 0 p ′ ( z 0 ) p ( z 0 ) = i l ,$
where:
$l ≥ m 2 a + 1 a when arg { p ( z 0 ) } = π 2$
and:
$l ≤ − m 2 a + 1 a when arg { p ( z 0 ) } = − π 2 ,$
where $p ( z 0 ) = ± i a$, $a > 0$.

## 2. Main Results

Theorem 1.
Let $λ 0 , λ , β , a ∈ R$ satisfy $λ ≥ 0$, $λ 0 a ≥ 0$, $0 < β ≤ 1$ and $| a | ≤ 1 β$. If q is analytic in $U$ with $q ( 0 ) = 1$ and satisfies:
$λ 0 ( q ( z ) ) a + λ q ( z ) + z q ′ ( z ) q ( z ) ≺ h ( z ) ( z ∈ U ) ,$
where:
$h ( z ) = λ 0 1 + z 1 − z a β + λ 1 + z 1 − z β + 2 β z 1 − z 2$
is (close-to-convex) univalent in $U$, then:
$arg { q ( z ) } < β π 2 ( z ∈ U ) .$
The bound β in (8) is sharp for the function q defined by:
$q ( z ) = 1 + z 1 − z β .$
Proof.
We choose:
$g ( z ) = 1 + z 1 − z β , θ ( w ) = λ 0 w a + λ w and φ ( w ) = 1 w$
in Lemma 1. Then, the function g is analytic and convex univalent in $U$ and:
$arg { g ( z ) } < β π 2 ( z ∈ U ) .$
It is clear that $φ$ and $θ$ are analytic in a domain $D$, which contains $g ( U )$ and $q ( U )$ with $φ ( w ) ≠ 0$ for $w ∈ g ( U )$. The function Q given by:
$Q ( z ) = z g ′ ( z ) φ ( g ( z ) ) = 2 β z 1 − z 2$
is univalent starlike. Further, we have:
$θ ( g ( z ) ) + Q ( z ) = λ 0 1 + z 1 − z a β + λ 1 + z 1 − z β + 2 β z 1 − z 2 = h ( z ) ,$
and so:
$z h ′ ( z ) Q ( z ) = θ ′ ( g ( z ) ) φ ( g ( z ) ) + z Q ′ ( z ) Q ( z ) = λ 0 a ( g ( z ) ) a + λ g ( z ) + z Q ′ ( z ) Q ( z ) .$
Furthermore, for $| a | ≤ 1 β$, we find that:
$arg ( g ( z ) ) a = | a | β π 2 ≤ π 2 ( z ∈ U ) .$
Therefore, it follows from (10)–(12) that:
$Re z h ′ ( z ) Q ( z ) > 0 ( z ∈ U ) .$
The other conditions of Lemma 1 are also satisfied. Hence, we conclude that:
$q ( z ) ≺ g ( z ) = 1 + z 1 − z β ( z ∈ U )$
and the function g is the best dominant of (6).
Furthermore, for the function q defined by (9), we have:
$λ 0 ( q ( z ) ) a + λ q ( z ) + z q ′ ( z ) q ( z ) = h ( z )$
and it follows that the bound $β$ in (8) is sharp. The proof of Theorem 1 is completed. □
Theorem 2.
Let $α > 0$, $0 < β < 1$ and $δ > 0$. If $f ∈ A ( p )$ satisfies $f ( z ) f ′ ( z ) ≠ 0$$( 0 < | z | < 1 )$ and:
$arg α 1 + z f ″ ( z ) f ′ ( z ) − z f ′ ( z ) f ( z ) + z f ′ ( z ) p f ( z ) δ < β π 2 ( z ∈ U ) ,$
then:
$arg z f ′ ( z ) f ( z ) < β π 2 δ ( z ∈ U ) .$
In particular, if $δ ≥ 1$, then f is p-valently strongly starlike of order $β δ$. The bound $β π 2$ in (13) is the largest number such that (14) holds true.
Proof.
One can see that the condition (13) is a generalization of the condition (4). For $f ∈ A ( p )$ satisfying $f ( z ) f ′ ( z ) ≠ 0$$( 0 < | z | < 1 )$, we define the function $p ( z )$ by:
$p ( z ) = z f ′ ( z ) p f ( z ) δ ( z ∈ U ) .$
Then, $p ( z )$ is analytic in $U$ with $p ( 0 ) = 1$. The condition (13) becomes:
$arg p ( z ) + α δ z p ′ ( z ) p ( z ) < β π 2 ( z ∈ U ) .$
Putting:
$λ 0 = 0 and λ = δ α$
in Theorem 1 and using (16), we find that if:
$α δ α 1 + z f ″ ( z ) f ′ ( z ) − z f ′ ( z ) f ( z ) + z f ′ ( z ) p f ( z ) δ = δ α p ( z ) + z p ′ ( z ) p ( z ) ≺ h ( z ) ,$
where:
$h ( z ) = δ α 1 + z 1 − z β + 2 β z 1 − z 2$
is (close-to-convex) univalent in $U$, then (14) is true.
Letting $0 < θ < π$ and $x = cot θ 2$, we deduce that:
$arg { h ( e i θ ) } = arg δ α 1 + e i θ 1 − e i θ β + 2 β e i θ 1 − e 2 i θ = arg δ α x β e β π i 2 + β i 2 x + 1 x = arctan δ x β sin β π 2 + α β 2 x + 1 x δ x β cos β π 2 ≥ β π 2 .$
Hence, in view of $h ( e − i θ ) = h ( e i θ ) ¯$, we deduce from (19) that $h ( U )$ contains the sector $| arg w | < β π 2$. Consequently, if $f ∈ A ( p )$ satisfies (13), then the subordination (17) holds true.
For the function f defined by:
$f ( z ) = exp p ∫ 0 z 1 t 1 + t 1 − t β δ d t ∈ A ( p ) ,$
we find after some computations that f satisfies (14) and:
$α δ α 1 + z f ″ ( z ) f ′ ( z ) − z f ′ ( z ) f ( z ) + z f ′ ( z ) p f ( z ) δ = h ( z ) ,$
which shows that the bound $β π 2$ in (13) is the largest number such that (14) holds true. The proof of Theorem 2 is completed. □
Theorem 3.
Let $δ > 0$ and $α ∈ C$. Assume that $− π 2 < φ = arg { α } ≤ 0$. If $f ∈ A ( p )$ satisfies $f ( z ) f ′ ( z ) ≠ 0$$( 0 < | z | < 1 )$ and:
$− π 2 + arctan | α | sin φ 2 δ + | α | cos φ < arg α 1 + z f ″ ( z ) f ′ ( z ) − z f ′ ( z ) f ( z ) + z f ′ ( z ) p f ( z ) δ < π 2 + φ$
for $z ∈ U$, then:
$arg z f ′ ( z ) f ( z ) < π 2 δ ( z ∈ U ) .$
In particular, if $δ ≥ 1$, then f is p-valently strongly starlike of order $1 δ$.
Proof.
Define the function $p ( z )$ by (15). Then, the condition (20) becomes:
$− π 2 + arctan | α | sin φ 2 δ + | α | cos φ < arg p ( z ) + α δ z p ′ ( z ) p ( z ) < π 2 + φ ( z ∈ U ) .$
We want to prove that:
$| arg { p ( z ) } | < π 2 ( z ∈ U ) .$
If there exists a point $z 0$$( | z 0 | < 1 )$ such that:
$| arg { p ( z ) } | < π 2 ( | z | < | z 0 | )$
and:
$| arg { p ( z 0 ) } | = π 2 ,$
then from Lemma 2, we have:
$z 0 p ′ ( z 0 ) p ( z 0 ) = i l ,$
where $p ( z 0 ) = ± a i$, $a > 0$ and:
$l ≥ m 2 a + 1 a when arg { p ( z 0 ) } = π 2$
and:
$l ≤ − m 2 a + 1 a when arg { p ( z 0 ) } = − π 2 .$
For the case $arg { p ( z 0 ) } = − π 2$, we have $l < 0$ and:
$arg p ( z 0 ) + α δ z 0 p ′ ( z 0 ) p ( z 0 ) = arg − a i + l α δ i = − π 2 + arg a − l α δ = − π 2 + arctan Im a − l α δ Re a − l α δ = − π 2 + arctan − l | α | sin φ a δ − l | α | cos φ ≤ − π 2 + arctan 1 2 a + 1 a | α | sin φ a δ + 1 2 a + 1 a | α | cos φ ≤ − π 2 + Q ( α , φ ) ,$
where:
$Q ( α , φ ) = max a > 0 arctan | α | ( a 2 + 1 ) sin φ ( 2 δ + | α | cos φ ) a 2 + | α | cos φ .$
The function:
$g ( a ) = | α | ( a 2 + 1 ) sin φ ( 2 δ + | α | cos φ ) a 2 + | α | cos φ , a > 0$
has a positive derivative:
$g ′ ( a ) = − 4 a δ | α | sin φ ( ( 2 δ + | α | cos φ ) a 2 + | α | cos φ ) 2 ≥ 0 for − π 2 < φ ≤ 0 ,$
hence:
$Q ( α , φ ) = lim a → ∞ arctan { g ( a ) } = arctan | α | sin φ 2 δ + | α | cos φ .$
Therefore, (23) becomes:
$arg p ( z 0 ) + α δ z p ′ ( z 0 ) p ( z 0 ) ≤ − π 2 + arctan | α | sin φ 2 δ + | α | cos φ ,$
$| arg { p ( z ) } | < π 2 ( z ∈ U ) .$
For the case $arg { p ( z 0 ) } = π 2$, applying the same method as the above, we have $l > 0$ and:
$arg p ( z 0 ) + α δ z 0 p ′ ( z 0 ) p ( z 0 ) = arg a i + l α δ i = π 2 + arg a + l α δ ≥ π 2 + φ .$
This contradicts (21). Now, the proof of Theorem 3 is completed. □
Remark 1.
For $φ = 0$ and $δ = 1$, Theorem 3 becomes the known result in [17] that every α-convex function is starlike.
Applying the same method as the above, we can prove the following theorem.
Theorem 4.
Let $δ > 0$ and $α ∈ C$. Assume that $0 ≤ φ = arg { α } < π 2$. If $f ∈ A ( p )$ satisfies $f ( z ) f ′ ( z ) ≠ 0$$( 0 < | z | < 1 )$ and:
$− π 2 + φ < arg α 1 + z f ″ ( z ) f ′ ( z ) − z f ′ ( z ) f ( z ) + z f ′ ( z ) p f ( z ) δ < π 2 + arctan | α | sin φ 2 δ + | α | cos φ$
for $z ∈ U$, then:
$arg z f ′ ( z ) f ( z ) < π 2 δ ( z ∈ U ) .$
In particular, if $δ ≥ 1$, then f is p-valently strongly starlike of order $1 δ$.
Theorem 5.
Theorem 5. Let $0 < α < 1$, $0 < β < 1$ and $β < δ ≤ 1$. If $f ∈ A ( p )$ satisfies $f ( z ) f ′ ( z ) ≠ 0$$( 0 < | z | < 1 )$ and:
$arg α 1 + z f ″ ( z ) f ′ ( z ) + ( 1 − α ) z f ′ ( z ) p f ( z ) δ < β π 2 ( z ∈ U ) ,$
then:
$arg z f ′ ( z ) f ( z ) < β π 2 δ ( z ∈ U ) ,$
or f is p-valently strongly starlike of order $β δ$. The bound $β π 2$ in (25) is the largest number such that (26) holds true.
Proof.
It is obvious that the condition (25) is a generalization of the condition (4). Defining the function $p ( z )$ by (15), the condition (25) becomes:
$arg p α ( p ( z ) ) 1 δ + ( 1 − α ) p ( z ) + α δ z p ′ ( z ) p ( z ) < β π 2 ( z ∈ U ) .$
Setting:
$a = 1 δ , λ 0 = p δ and λ = δ ( 1 − α ) α$
in Theorem 1 and using (27), we see that if:
$α δ α 1 + z f ″ ( z ) f ′ ( z ) + ( 1 − α ) z f ′ ( z ) p f ( z ) δ = p δ ( p ( z ) ) 1 δ + δ ( 1 − α ) α p ( z ) + z p ′ ( z ) p ( z ) ≺ h ( z ) ,$
where:
$h ( z ) = p δ 1 + z 1 − z β δ + δ ( 1 − α ) α 1 + z 1 − z β + 2 β z 1 − z 2$
is (close-to-convex) univalent in $U$, then (26) holds true.
Letting $0 < θ < π$ and $x = cot θ 2$, we have:
$h ( e i θ ) = p δ x β δ e β π i 2 δ + δ ( 1 − α ) α x β e β π i 2 + β i 2 x + 1 x$
and:
$arg h ( e i θ ) = arctan p δ x β δ sin β π 2 δ + δ ( 1 − α ) α x β sin β π 2 + β 2 x + 1 x p δ x β δ cos β π 2 δ + δ ( 1 − α ) α x β cos β π 2 .$
For $x > 0$, $0 < α < 1$ and $0 < β δ < 1$, we deduce that:
$arg h ( e i θ ) ≥ arctan p δ x β δ sin β π 2 δ + δ ( 1 − α ) α x β sin β π 2 p δ x β δ cos β π 2 δ + δ ( 1 − α ) α x β cos β π 2 = arctan tan β π 2 p δ x β δ cos β π 2 δ tan β π 2 δ cot β π 2 + δ ( 1 − α ) α x β cos β π 2 p δ x β δ cos β π 2 δ + δ ( 1 − α ) α x β cos β π 2 ≥ β π 2 ,$
since:
$tan β π 2 δ cot β π 2 ≥ tan β π 2 cot β π 2 = 1 for 1 δ ≥ 1 .$
In view of the proof of Theorem 2, we find from (30) that $h ( U )$ contains the sector $| arg w | < β π 2$. Hence, if $f ∈ A ( p )$ satisfies (25), then the subordination (28) holds true.
The sharpness part of the proof is similar to that in the proof of Theorem 2, and so, we omit it. The proof of Theorem 5 is completed. □

## Author Contributions

All authors contributed equally.

## Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11571299).

## Acknowledgments

The authors would like to express sincere thanks to the referees for careful reading and suggestions, which helped us to improve the paper.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Brannan, D.A.; Kirwan, W.E. On some classes of bounded univalent functions. J. Lond. Math. Soc. 1969, 1, 431–443. [Google Scholar] [CrossRef]
2. Ali, M.F.; Vasudevarao, A. Logarithmic coefficients of some close-to-convex functions. Bull. Aust. Math. Soc. 2017, 95, 228–237. [Google Scholar] [CrossRef]
3. Ali, M.F.; Vasudevarao, A. Coefficient inequalities and Yamashita’s conjecture for some classes of analytic functions. J. Aust. Math. Soc. 2016, 100, 1–20. [Google Scholar] [CrossRef]
4. Baricz, Á.; Szász, R. Close-to-convexity of some special functions and their derivatives. Bull. Malaysian Math. Sci. Soc. 2016, 39, 427–437. [Google Scholar] [CrossRef]
5. Gangadharan, A.; Ravichandran, V. Radii of convexity and strong starlikeness for some classes of analytic functions. J. Math. Anal. Appl. 1997, 211, 303–313. [Google Scholar] [CrossRef]
6. Liu, J.-L. Notes on Jung-Kim-Srivastava integral operator. J. Math. Anal. Appl. 2004, 294, 96–103. [Google Scholar] [CrossRef]
7. Nunokawa, M.; Owa, S.; Saitoh, H.; Ikeda, A.; Koike, N. Some results for strongly starlike functions. J. Math. Anal. Appl. 1997, 212, 98–106. [Google Scholar] [CrossRef]
8. Nunokawa, M.; Sokól, J.; Trabka-Wieclaw, K. On the order of strongly starlikeness in some classes of starlike functions. Acta Math. Hung. 2015, 145, 142–149. [Google Scholar] [CrossRef]
9. Nunokawa, M.; Thomas, D.K. On convex and starlike functions in a sector. J. Aust. Math. Soc. Ser. A 1996, 60, 363–368. [Google Scholar] [CrossRef]
10. Obradorić, M.; Owa, S. Some sufficient conditions for strongly starlikeness. Int. J. Math. Math. Sci. 2000, 24, 643–647. [Google Scholar] [CrossRef] [Green Version]
11. Owa, S.; Srivastava, H.M.; Hayami, T.; Kuroki, K. A new general idea for starlike and convex functions. Tamkang J. Math. 2016, 47, 445–454. [Google Scholar]
12. Ponnusamy, S.; Singh, V. Criteria for strongly starlike functions. Complex Var. Theory Appl. 1997, 34, 267–291. [Google Scholar] [CrossRef]
13. Prajapat, J.K.; Raina, R.K.; Srivastava, H.M. Some inclusion properties for certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operator. Integr. Transforms Spec. Funct. 2007, 18, 639–651. [Google Scholar] [CrossRef]
14. Shiraishi, H.; Owa, S.; Srivastava, H.M. Sufficient conditions for strongly Carathéodory functions. Comput. Math. Appl. 2011, 62, 2978–2987. [Google Scholar] [CrossRef] [Green Version]
15. Srivastava, H.M.; Yang, D.G.; Xu, N.E. Subordinations for multivalent analytic functions associated with the Dziok-Srivastava operator. Integral Transforms Spec. Funct. 2009, 20, 581–606. [Google Scholar] [CrossRef]
16. Yang, D.-G.; Liu, J.-L. Some subclasses of meromorphic and multivalent functions. Ann. Polon. Math. 2014, 111, 73–88. [Google Scholar] [CrossRef]
17. Mocanu, P.T. Une propriété de convexité géneralisée dans la theórie de la représentation conforme. Mathematica 1969, 11, 127–133. [Google Scholar]
18. Miller, S.S.; Mocanu, P.T. On some classes of first-order differential subordinations. Michigan Math. J. 1985, 32, 185–195. [Google Scholar] [CrossRef]
19. Nunokawa, M. On properties of non-Carathéodory function. Proc. Jpn. Acad. Ser. A 1992, 68, 152–153. [Google Scholar] [CrossRef]

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Yuan, Y.; Srivastava, R.; Liu, J.-L. The Order of Strongly Starlikeness of the Generalized α-Convex Functions. Symmetry 2019, 11, 76. https://doi.org/10.3390/sym11010076

AMA Style

Yuan Y, Srivastava R, Liu J-L. The Order of Strongly Starlikeness of the Generalized α-Convex Functions. Symmetry. 2019; 11(1):76. https://doi.org/10.3390/sym11010076

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Yuan, Yuan, Rekha Srivastava, and Jin-Lin Liu. 2019. "The Order of Strongly Starlikeness of the Generalized α-Convex Functions" Symmetry 11, no. 1: 76. https://doi.org/10.3390/sym11010076

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