Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300 DOI 10.1186/s13660-015-0818-x
RESEARCH
Open Access
On some geometric properties of multivalent functions Mamoru Nunokawa1 and Janusz Sokół2* *
Correspondence:
[email protected] Department of Mathematics, Rzeszów University of Technology, ´ Al. Powstanców Warszawy 12, Rzeszów, 35-959, Poland Full list of author information is available at the end of the article 2
Abstract We prove some new sufficient conditions for a function to be p-valent or p-valently starlike in the unit disc. MSC: Primary 30C45; secondary 30C80 Keywords: analytic functions; p-valent functions; p-valently starlike
1 Introduction Let Ap be the class of analytic functions of the form f (z) = zp +
∞
ap+n zp+n
(z ∈ D).
(.)
n=
A function f (z) which is analytic in a domain D ⊂ C is called p-valent in D if for every complex number w, the equation f (z) = w has at most p roots in D, and there will be a complex number w such that the equation f (z) = w has exactly p roots in D. Further, a function f ∈ Ap , p ∈ N \ {}, is said to be p-valently starlike of order α, ≤ α < p, if zf (z) >α Re f (z)
(z ∈ D).
The class of all such functions is usually denoted by Sp∗ (α). For p = , we receive the wellknown class of normalized starlike univalent functions S ∗ (α) of order α, Sp∗ () = Sp∗ . If zf (z) ∈ S ∗ (α), then f (z) is said to be p-valently convex of order α, ≤ α < p. The class of all such functions is usually denoted by Cp (α). For p = , we receive the well-known class of normalized convex univalent functions C (α) of order α, Cp () = Cp . The well-known Noshiro-Warschawski univalence condition (see [] and []) indicates that if f (z) is analytic in a convex domain D ⊂ C and Re eiθ f (z) > (z ∈ D), where θ is a real number, then f (z) is univalent in D. In [] Ozaki extended the above result by showing that if f (z) of the form (.) is analytic in a convex domain D and for some real © 2015 Nunokawa and Sokół. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300
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θ we have Re eiθ f (p) (z) > ,
|z| < ,
then f (z) is at most p-valent in D. Applying Ozaki’s theorem, we find that if f (z) ∈ Ap and Re f (p) (z) > (z ∈ D), then f (z) is at most p-valent in |z| < . In [] it was proved that if f (z) ∈ Ap , p ≥ , and π , argf (p) (z) <
|z| < ,
then f (z) is at most p-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 arg p(z) < πα
for |z| < |z |
and arg p(z ) = πα for some < α < , then we have z p (z ) = isα, p(z ) where s≥
a+ a
πα when arg p(z ) =
and s≤–
a+ a
πα when arg p(z ) = – ,
where /α p(z ) = ±ia, and a > .
Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300
Lemma . [] Let p be a positive integer. If f (z) = zp + it satisfies
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∞
n=p+ an z
n
is analytic in D and if
zf (p) (z) Re (p–) > , f (z)
z ∈ D,
(.)
then f (z) is p-valently starlike in D and 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
n g(z) = ∞ n=p–k+ bn z (bp–k+ = ) that satisfies zf (k) (z) > , Re 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 ∈ Ap and suppose that (p+) f (z)< α(β )(p!),
|z| < ,
(.)
where β = . · · · is the positive root of the equation β +
tan– β = , π
, f (z)
|z| < ,
(.)
or f (z) is p-valently starlike in |z| < . Proof From the hypothesis (.), we have z (p) f (z) – p! = f (p+) (t) dt
|z|
f (p+) (t)|dt| < p!
≤
|z|
α(β )|dt|
= p!α(β )|z| < p!α(β ). This shows that (p) arg f (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 point z , |z | < , such that arg q(z) < πβ
for |z| < |z |
and arg q(z ) = πβ , then by Lemma . we have z q (z ) = iβ k, q(z ) where πβ k ≥ when arg q(z ) =
(.)
πβ k ≤ – when arg q(z ) = – .
(.)
and
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For the case (.), we have (p) (p) f (z) arg f (z) = arg p! = arg q(z ) + z q (z ) z q (z ) = arg 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 arg f (p) (z) ≤ – sin– α(β ). This also contradicts (.) and, therefore, it shows that (p–) (z) πβ arg f < , p!z
|z| < .
(.)
Applying (.) and (.), we have (p) z arg zf (z) = arg f (p) (z) + arg (p–) (p–) f (z) f (z) (p) z ≤ arg f (z) + arg (p–) f (z) πβ < sin– α(β ) + π = β + tan– β π π = . This shows that zf (p) (z) > , Re (p–) f (z)
|z| < ,
and by Lemma . we have Re
zf (z) > , f (z)
|z| < ,
or f (z) is p-valently starlike in |z| < .
Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300
Remark Note that if m(x) = x +
π
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tan– x, then
m(.) = . · · · ,
m(.) = . · · · .
Hence, if β +
tan– β = , π
then β = . · · · . Moreover, . · · · < sin– α(β ) =
( – β )π π β + tan– β = < . · · · π
and π β + tan– β = . · · · . α(β ) = sin π For p = , Theorem . becomes the following corollary which extends the result contained in Lemma .. Corollary . Let f ∈ A() and suppose that f (z) < α(β ),
|z| < ,
(.)
where β = . · · · is the positive root of the equation β +
tan– β = , π
< β < ,
(.)
and ( – β )π α(β ) = sin = . · · · . Then we have zf (z) > , Re f (z)
|z| < ,
(.)
or f (z) is starlike univalent in |z| < . An analytic function f (z) is said to be typically real if the inequality ImzImf (z) ≥ holds for all z ∈ D. From the definition of a typically real function it follows that z ∈ D+ ⇔ f (z) ∈ C+ and z ∈ D– ⇔ f (z) ∈ C– . The symbols D+ , D– , C+ , C– stand for the following sets: the upper and the lower halves of the disk D, the upper half-plane and the lower half-plane, respectively.
Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300
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Theorem . Let f (z) ∈ Ap and suppose that (p) arg zf (z) < π + tan– – |z| , g(z) + |z|
z ∈ D,
(.)
where g(z) is univalent starlike in D and the functions zg (z) g(z)
and
z( zf
(p–) (z)
g(z)
)
zf (p–) (z) g(z)
(.)
are typically real in D. Then we have (p–) (z) π arg zf < , g(z)
z ∈ D.
(.)
Proof Let us put q(z) =
zf (p–) (z) , p!g(z)
q() = .
Then it follows that zf (p) (z) zg (z) zq (z) = p!q(z) + . g(z) g(z) q(z)
(.)
From the hypothesis zg (z) zq (z) + g(z) q(z) is typically real in D. If there exists a point z , |z | < , such that arg q(z) < π
for |z| < |z |
and arg q(z ) = π ,
q(z ) = ±ia, and a > ,
then by Lemma . we have z q (z ) = is, q(z )
(.)
where s≥ a+ a
π when arg q(z ) =
and s≤–
a+ a
π when arg q(z ) = – .
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For the case π arg q(z ) = ,
q(z ) = ia, a > ,
we have z q (z ) = is, q(z )
s≥ a+ > , a
(.)
hence Im{z } >
(.)
because zq (z)/q(z) is typically real. Therefore, (.) yields that
z g (z ) Im > , g(z )
(.)
because zg (z)/g(z) is typically real. Moreover, z g (z ) + |z | – |z | ≤ Re ≤ + |z | g(z ) – |z |
(.)
because g(z) is a univalent starlike function, see []. Applying (.), (.) and (.) in (.), we have (p) z g (z ) z q (z ) z f (z ) = arg q(z ) + arg + arg g(z ) g(z ) q(z ) z g (z ) + is = arg q(z ) + arg g(z ) ≥
– |z | π + tan– . + |z |
This contradicts (.), and for the case π arg q(z ) = – , applying the same method as above, we have (p) z g (z ) z q (z ) z f (z ) arg = arg q(z ) + arg + g(z ) g(z ) q(z ) – |z | π + tan– . ≤– + |z | This also contradicts (.) and, therefore, it shows that (.) holds. Corollary . Let f (z) ∈ Ap and all the coefficients of f (z) are real and suppose that (p) arg zf (z) < π + tan– – |z| , g(z) + |z|
z ∈ D,
Nunokawa and Sokół Journal of Inequalities and Applications (2015) 2015:300
where g(z) is univalent starlike and typically real in D. Then we have (p–) (z) π arg zf < , g(z)
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. Powstanców 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. 1 2(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. A 2, 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. A 124(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. TMA 71, 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. TMA 71, 6323-6342 (2009) 10. Goodman, AW: Univalent Functions, Vols. I and II. Mariner Publishing Co., Tampa (1983)
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