A Class of Locally Univalent Functions Defined by a Differential Inequality R. Fournier∗
S. Ponnusamy†
CRM-3204 October 2005
∗ D´ epartement de Math´ ematiques et Centre de recherches math´ ematiques, Universit´ e de Montr´ eal, C.P. 6128, succ. Centre-ville, Montr´ eal, Qc H3C 3J7, Canada;
[email protected] † Department of Mathematics, Indian Institute of Technology Madras, Chennai—600 036, India;
[email protected]
Abstract We study the range of parameters λ and µ such that any function f (z) analytic for |z| < 1 with µ+1 z f (z) 0 6= 0 and f (z) − 1 < λ ≤ 1, |z| < 1, z f (z) is starlike or spirallike. Dedicated to the memory of Vikramaditya Singh. Keywords and phrases. Differential inequalities, starlike functions, analytic functions 2000 Mathematical Subject Classification. 30C45
1
Introduction and Statement of the Results
Let H(D) denote the space of functions analytic in the unit disc D := {z | |z| < 1} of the complex plane C; here we think of H(D) as a topological vector space endowed with the topology of uniform convergence over compact subsets of D. Further, let A := {f ∈ H(D) | f (0) = f 0 (0) − 1 = 0}; we say that a function f ∈ A is starlike if f is univalent in D and f (D) is a domain with the property that the segment [0, w] := {tw | 0 ≤ t ≤ 1} ⊂ f (D) for any w ∈ f (D). We denote by St the class of starlike functions. More generally, a function f ∈ A is called spirallike if f is univalent in D and the following property holds: the unique logarithmic spiral linking the origin and any w ∈ f (D) is also included in f (D). We denote by Sp the class of spirallike functions and clearly St ⊂ Sp. There exists a huge literature on these classical classes of univalent functions and we refer the reader to [2] concerning analytical definitions of St and Sp. We shall be concerned in this paper with the classes ) ( µ+1 f (z) z 0 6= 0 and f (z) − 1 < λ, z ∈ D U (λ, µ) := f ∈ H(D) z f (z) and U (λ) := U (λ, 1). It is known ([1, 7, 11]) that the functions in U (λ) are univalent if 0 ≤ λ ≤ 1 but not necessarily univalent if λ > 1. It is also known [3] that U (1, −1) 6⊂ St and V. Singh [15] gave an estimate for the radius of starlikeness (which is surprisingly close to unity) of U (1, −1). More recently, Obradovi´c, Ponnusamy and Singh ([8, 10]) proved that U (λ, µ) ⊂ St if µ < 1
1−µ . and 0 ≤ λ ≤ p (1 − µ)2 + µ2
Our main results are the following: Theorem 1. Let µ ∈ C with Re(µ) < 1. Then U (λ, µ) ⊂ St
|1 − µ|
iff
0≤λ≤ p
iff
|1 − µ| 0 ≤ λ ≤ min 1, . |µ|
|1 − µ|2 + |µ|2
.
Theorem 2. Let µ ∈ C with Re(µ) < 1. Then U (λ, µ) ⊂ Sp
Clearly U (1, µ) ⊂ St iff µ = 0 and U (1, µ) ⊂ Sp iff Re(µ) ≤ 21 .
2
Some Lemmas
A finite Blaschke product is a function of the type b(z) = eiγ
m Y z − aj , 1 −a ¯j z j=1
{aj }nj=1 ⊂ D,
γ ∈ R.
The following result will be useful for several of our sharpness arguments Lemma 1. Given ϕ and ψ in R, there exists a sequence {bn } of finite Blaschke products such that bn (1) = eiϕ , bn (0) = 0 and bn (z) → eiψ z in the sense of convergence in H(D). This surprising (at first sight!) result has been published first in [3] and later in [12]; the products bn consist in fact of at most two factors. We wish to point out however that due to previous work of Maurice Heins ([5]) and more recent work on interpolation by Blaschke products (see [6] for quick references) a stronger (in some sense!) result is indeed available: Lemma 10 . There exists an infinite sequence {Wn } of finite Blaschke products with the following property: given a m function w ∈ H(D) with w(D) ⊆ D and two sets of nodes {ϕk }m k=1 and {ψk }k=1 in R where the ϕk ’s are assumed to be pairwise distinct (mod 2π), there exists a subsequence {Wnj } of {Wn } such that Wnj (eiϕk ) = eiψk ,
1 ≤ k ≤ m, j = 1, 2, . . .
and lim Wnj = w
j→∞
1
in H(D).
We recall that the Hadamard product f ? g of two power series f (z) := in H(D) is the power series defined by ∞ X f ? g(z) := an (f )an (g)z n .
P∞
n=0
an (f )z n and g(z) :=
P∞
n=0
an (g)z n
n=0
It is clear that f ? g is also a member of H(D). Our next Lemma is essentially due to Ruscheweyh [13]: P∞ 1−c n−1 Lemma 2. Let c ∈ C with Re(c) < 1 and Fc (z) := n=1 n−c z ∈ H(D). Then sup |f ? Fc (z)| ≤ sup |f (z)|, z∈D
for any f ∈ H(D).
z∈D
Let us finally consider j ≥ 0 and Bj = {w ∈ H(D) | |w(z)| ≤ |z|j , z ∈ D}. Clearly Bj is a subspace of H(D) and a topological space of its own. We shall also need Lemma 3. Let θ ∈ R and Re(c) < j. Then the functional I(w) =
∞ X ak (w) k=j
k−c
eikθ ,
w(z) =
∞ X
ak (w)z k ∈ Bj
k=j
is well defined and continuous over Bj . A short proof of Lemma 3 can be found in [4, pp. 285–286].
3
Proof of Theorem 1
Let f ∈ U (λ, µ) with Re(µ) < 1. Then µ+1 z f 0 (z) − 1 = λw(z), f (z)
|z| < 1, for some w ∈ B1 ,
(3.1)
and upon integrating (3.1) we obtain
f (z) z
−µ =1−λ
µ w(z) ? zFµ (z) + Kz µ , 1−µ
|z| < 1
for some complex constant K. Since µ is not a positive integer and because of the analyticity of conclude that K = 0. It follows then from (3.1) and (3.2) that zf 0 (z) 1 + λw(z) = . f (z) 1 − λµ/(1 − µ)zFµ ? w(z)
(3.2) f (z) −µ , z
we must
(3.3)
By Lemma 2 and the maximum principle,
Re
zf 0 (z) f ∈ U (λ, µ), z ∈ D f (z) ⊆
Re
1 + λeiϕ ϕ, ψ ∈ R . (3.4) 1 + λ|µ/(1 − µ)|eiψ
We indeed have equality in (??): by Lemma 1, there exists a sequence {wn } ⊂ B1 of finite Blaschke products such that, given ϕ, ψ ∈ R, wn (1) = eiϕ and wn (z) → ei(π+ψ−arg(µ/(1−µ)) z in H(D). Defining {fn } ⊂ U (λ, µ) by
fn0 (z)
z fn (z)
µ+1 − 1 = λwn (z)
µ (here we may assume λ| 1−µ | < 1) we see that, by (??) and Lemma 3,
fn0 (1) 1 + λeiϕ = n→∞ fn (1) 1 + λ|µ/(1 − µ)|eiψ lim
2
and equality holds in (??). Now by the well-known criterion for starlikeness ([2, pp. 40–46]) we have 1 + λeiϕ U (λ, µ) ⊂ St ⇐⇒ Re ≥ 0 for all real ϕ, ψ. 1 + λ|µ/(1 − µ)|eiψ µ This means that the hypothesis λ| 1−µ | < 1 was necessary and we get
µ ≤π U (λ, µ) ⊂ St ⇐⇒ arcsin(λ) + arcsin λ 1 − µ 2 and this last inequality is seen to be satisfied if and only if 0 ≤ λ ≤ √
|1−µ| |1−µ|2 +|µ|2
.
false if µ is assumed to be an integer ≥ 1. Given ψ ∈ R, (ψ 6= 0 It is remarkable that Theorem 1 is drastically P∞ mod 2π), 0 < λ < µ1 and w ∈ Bµ+1 and w(z) = k=µ+1 ak (w)z k to be specified later, let us define −1/µ ∞ X a (w) k z k − (1 − λµ)eiψ z µ . f (z) = z 1 − λµ k−µ
k=µ+1
Since Z 1 ∞ X ak (w) k w(tz) iψ µ iψ µ λµ λµ z + (1 − λµ)e z dt + (1 − λµ)e z = 1+µ 0 t k=µ+1 k − µ Z 1 |w(tz)| dt + (1 − λµ)|z|µ ≤ λµ t1+µ 0 ≤ λµ|z|1+µ + (1 − λµ)|z|µ < 1, the function f is well-defined and mild computations yield µ+1 z 0 f (z) = 1 + λw(z) f (z) and
(3.5)
1 + λw(z) zf 0 (z) P∞ = . f (z) 1 − λµ k=µ+1 (ak (w)/(k − µ))z k − (1 − λµ)eiψ z µ
(3.6)
It follows from (??) that f ∈ U (λ, µ) for any choice of w ∈ Bµ+1 . Now given a real number ϕ, we choose a sequence {wn } ⊂ Bµ+1 of finite Blaschke products such that wn (1) = eiϕ
and
lim wn (z) = eiψ z µ+1
n→∞
in H(D).
This is of course possible because of Lemma 1 and by Lemma 3 and (??), we have for the corresponding functions fn ∈ U (λ, µ), f 0 (1) 1 + λeiϕ lim n = . n→∞ fn (1) 1 − eiψ This evidently means that inf
|z| 12 . We finally remark that according to (3.2) we have for any f ∈ U (1, µ) −1/µ
Z
f (z) = (−µ)
z
1 + w(ζ) ζ − Im(µ)i−1 ζ − Re(µ) dζ
1/(− Re(µ)−Im(µ)i)
0
which means that if Re(µ) < 0, the functions in U (1, µ) are not only spirallike but also Bazileviˇc functions (see [14] for details).
5
Concluding Remarks
Using the methods developed in this paper, it is possible to determine sharp intervals for λ and/or sharp regions of variability for µ for U (λ, µ) to be included in various classical classes of univalent functions such as the class of functions of bounded turning or the class of strongly starlike functions or the class of starlike functions of a given order. The following results (given here without proof, see [9]) offer an alternative to the fact that for arbitrarily small values of λ we have U (λ, 1) 6⊂ St. We have P∞ Theorem 3. Any function f (z) := z + n=2 an (f )z n ∈ A satisfying 2 −|a (f )| + p2 − |a (f )|2 z 0 2 2 − 1 < , |z| < 1, f (z) f (z) 2 belongs to St. Moreover, there exists a non-starlike function f in U (1) such that p 2 −|a2 (f )| + 2 − |a2 (f )|2 z 0 0< < sup f (z) − 1 ≤ 1 − |a2 (f )|. 2 f (z) |z|
