General Mathematics Vol. 18, No. 3 (2010), 51–58
Starlikeness of analytic maps satisfying a
differential inequality 1
Sukhwinder Singh, Sushma Gupta, Sukhjit Singh
Abstract
In the present note, the authors present a criterion for starlikeness of
analytic maps satisfying a differential inequality in the open unit disc E =
{z : |z| < 1} and claim that their result unifies a number of previously
known results in this direction.
2010 Mathematics Subject Classification: Primary 30C80, Secondary
30C45.
Key words and phrases: Analytic function, Starlike function, α-convex
function.
1
Introduction
Let A be the class of functions f , analytic in E = {z : |z| < 1} and normalized
by the conditions f (0) = f 0 (0) − 1 = 0. Denote by S ∗ (α), the class of starlike
functions of order α, which is analytically defined as follows:
zf 0 (z)
∗
S (α) = f ∈ A : <
> α, z ∈ E ,
f (z)
where α is a real number such that 0 ≤ α < 1.
We write S ∗ = S ∗ (0). Therefore S ∗ is the class of univalent starlike functions (w.r.t. the origin).
1
Received 23 February, 2009
Accepted for publication (in revised form) 17 March, 2009
51
52
S. Singh, S. Gupta, S. Singh
Obtaining different criteria for starlikeness of an analytic function has always been a subject of interest e.g. Miller, Mocanu and Reade [5] studied the
class of α-convex functions and proved that if a function f ∈ A satisfies the
differential inequality
zf 0 (z)
zf 00 (z)
< (1 − α)
> 0, z ∈ E,
+α 1+ 0
f (z)
f (z)
where α is any real number, then f is starlike in E. Lewandowski et al.[2]
proved that for a function f ∈ A, the differential inequality
0
zf (z) z 2 f 00 (z)
+
<
> 0, z ∈ E,
f (z)
f 0 (z)
ensures membership for f in the class S ∗ . For more such results, we refer the
reader to [3], [6], [7] and [8].
In the present paper, we generalize these sufficient conditions and obtain
an interesting criterion for starlikeness. In Section 4, we show that some wellknown results follow as corollaries to our result.
2
Preliminaries
We shall need the following lemma of Miller and Mocanu [4] to prove our
result.
Lemma 1 Let Ω be a set in the complex plane C and let ψ : C2 × E → C. For
u = u1 + iu2 , v = v1 + iv2 , assume that ψ satisfies the condition ψ(iu2 , v1 ; z) ∈
/
2
Ω, for all u2 , v1 ∈ R, with v1 ≤ −(1 + u2 )/2 and for all z ∈ E. If the function
p, p(z) = 1 + p1 z + p2 z 2 + . . ., is analytic in E and if ψ(p(z), zp0 (z); z) ∈ Ω,
then < p(z) > 0 in E.
3
Main Theorem
Theorem 2 Let α, α ≥ 0, λ, 0 ≤ λ < 1, and β, 0 ≤ β ≤ 1, be given real
numbers.
(i) For 1/2 ≤ λ < 1, if a function f ∈ A, f (z)
z 6= 0 in E, satisfies
(1)
αzf 00 (z)
zf 0 (z)
zf 00 (z)
zf 0 (z)
1+
+ αβ 1 −
1+ 0
> M (α, β, λ),
<
f (z)
f 0 (z)
f (z)
f (z)
53
Starlikeness of analytic maps satisfying ...
then f ∈ S ∗ (λ).
(ii) For 0 ≤ λ < 1/2, let a function f ∈ A,
(a)
(2)
zf 0 (z)
<
f (z)
αzf 00 (z)
1+
f 0 (z)
f (z)
z
zf 0 (z)
+ αβ 1 −
f (z)
6= 0 in E, satisfy
zf 00 (z)
1+ 0
f (z)
> M (α, β, λ),
> N (α, β, λ),
whenever
β(2λ − 1 − 3λ3 + 2λ4 ) + (3 − 2λ)λ3 ≥ 0,
(3)
and
(b)
(4)
zf 0 (z)
<
f (z)
αzf 00 (z)
1+
f 0 (z)
zf 0 (z)
+ αβ 1 −
f (z)
zf 00 (z)
1+ 0
f (z)
whenever
β(2λ − 1 − 3λ3 + 2λ4 ) + (3 − 2λ)λ3 ≤ 0.
(5)
Then f ∈ S ∗ (λ). Here
(6) M (α, β, λ) = [1 − α(1 − β)]λ + α(1 − β)λ2 −
α(1 − β)(1 − λ) αβ(1 − λ)
−
,
2
2λ
and
N (α, β, λ) = [1 − α(1 − β)]λ + α(1 − β)λ2 −
(7)
−
α(1 − β)(1 − λ)
2
p
α
[2 βλ(1 − 2λ)(1 − β)(3 − 2λ)+βλ−λ2 (1−β)(3−2λ)].
2(1 − λ)
Proof. Define a function p by
(8)
zf 0 (z)
= λ + (1 − λ)p(z).
f (z)
Then p is analytic in E and p(0) = 1. A simple calculation yields
54
S. Singh, S. Gupta, S. Singh
zf 0 (z)
f (z)
1+
αzf 00 (z)
f 0 (z)
+ αβ 1 −
zf 0 (z)
f (z)
1+
zf 00 (z)
f 0 (z)
= (1 − α + αβ)[λ + (1 − λ)p(z)] + α(1 − β)[λ + (1 − λ)p(z)]2
+α(1 − β)(1 − λ)zp0 (z) + αβ
(1 − λ)zp0 (z)
λ + (1 − λ)p(z)
= ψ(p(z), zp0 (z); z)
(9)
where,
ψ(u, v; z) = (1 − α + αβ)[λ + (1 − λ)u] + α(1 − β)[λ + (1 − λ)u]2
+α(1 − β)(1 − λ)v + αβ
(1 − λ)v
λ + (1 − λ)u
Let u = u1 + iu2 , v = v1 + iv2 , where u1 , u2 , v1 , v2 are all real with v1 ≤
−(1 + u22 )/2. Then, we have
< ψ(iu2 , v1 ; z)
= (1 − α + αβ)λ + α(1 − β)[λ2 − (1 − λ)2 u22 ] +
+α(1 − β)(1 − λ)v1 + αβ
λ2
λ(1 − λ)v1
+ (1 − λ)2 u22
≤ (1 − α + αβ)λ + α(1 − β)[λ2 − (1 − λ)2 u22 ] +
λ(1 − λ)(1 + u22 )
α(1 − β)(1 − λ)(1 + u22 )
− αβ
−
2
2(λ2 + (1 − λ)2 u22 )
α(1 − β)(1 − λ)
= (1 − α + αβ)λ + α(1 − β)λ2 −
−
2
λ(1 − λ)(1 + u22 )
3
−α(1 − β)(1 − λ)
− λ u22 − αβ
2
2(λ2 + (1 − λ)2 u22 )
α(1 − β)(1 − λ)
= (1 − α + αβ)λ + α(1 − β)λ2 −
−
2 λ(1 − λ)(1 + t)
3
−α(1 − β)(1 − λ)
− λ t − αβ
2
2(λ2 + (1 − λ)2 t)
= φ(t)
(say), where u22 = t
(10)≤ max φ(t).
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Starlikeness of analytic maps satisfying ...
Writing
α(1 − β)(1 − λ)
= a,
2
3
−λ =b
(1 − β)(1 − λ)
2
(1 − α + αβ)λ + α(1 − β)λ2 −
and
λ
1−λ
= c, we have
αβc
φ(t) = a − αbt −
2
1+t
c2 + t
.
Clearly, φ(t) is continuous at t = 0. A simple calculation gives
αβc
φ (t) = −αb −
2
0
c2 − 1
(c2 + t)2
.
λ
Case (i). When 1/2 ≤ λ < 1, then c = 1−λ
≥ 1. Since α ≥ 0, 0 ≤ β ≤ 1,
0
therefore, b > 0. Hence, φ (t) ≤ 0 which implies that φ is a decreasing function
of t (≥ 0). Thus
max φ(t) = φ(0)
(11)
= M (α, β, λ).
Let
Ω = {w : < w > M (α, β, λ)}.
Then from (1) and (9), we have ψ(p(z), zp0 (z); z) ∈ Ω for all z ∈ E, but
ψ(iu2 , v1 ; z) ∈
/ Ω, in view of (10) and (11). Therefore, by Lemma 1 and (8),
we conclude that f ∈ S ∗ (λ).
λ
Case (ii). When 0 ≤ λ < 1/2, we get c = 1−λ
< 1. Now, φ0 (t) = 0 implies
αβc
−αb −
2
c2 − 1
(c2 + t)2
=0
which gives
βc(1 − c2 )
.
2b
q
2)
2
= t1 and −c + βc(1−c
= t2 , we observe that
2b
t = −c2 ±
q
2)
Writing
− βc(1−c
2b
t1 < 0 and also, t1 < t2 .
−c2
r
56
S. Singh, S. Gupta, S. Singh
Subcase (i). When t2 < 0, i.e. when (3) holds true. In that case t1 and t2
both are negative. (Here, t is positive.) It can be easily verified that
φ0 (t) = −
α
(t − t1 )(t − t2 ) < 0.
2(c2 + t)2
Thus φ is a decreasing function of t and again
max φ(t) = φ(0)
(12)
= M (α, β, λ).
Proceeding as in case (i), we obtain the required result.
Subcase (ii). When t2 > 0, i.e. when (5) holds true. In that case, φ is a
increasing function of t, t ≥ 0, and therefore,
max φ(t) = φ(t2 )
(13)
= N (α, β, λ).
Let
Ω = {w : < w > N (α, β, λ)}.
Then from (4) and (9), we have ψ(p(z), zp0 (z); z) ∈ Ω for all z ∈ E, but
ψ(iu2 , v1 ; z) ∈
/ Ω, in view of (10) and (13). Result now follows by Lemma 1.
4
Applications to Univalent Functions
In this section, we apply Theorem 2 and obtain certain well-known criteria for
starlikeness of an analytic function.
Writing β = 1 in Theorem 2, we obtain the following result of Fukui [1]
for the class of α-convex functions.
Corollary 3 Let α, α ≥ 0 be a given real number. For all z ∈ E, let a
function f ∈ A satisfy
(
αλ
> λ − 2(1−λ)
, 0 ≤ λ < 1/2,
zf 0 (z)
zf 00 (z)
< (1 − α)
+α 1+ 0
α(1−λ)
f (z)
f (z)
> λ − 2λ , 1/2 ≤ λ < 1.
Then f ∈ S ∗ (λ).
The case, when we write β = 0 in Theorem 2, gives the following result of
Ravichandran et al. [9].
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Starlikeness of analytic maps satisfying ...
Corollary 4 Let α, α ≥ 0 be a given real number. For a real number λ, 0 ≤
λ < 1, and for all z ∈ E, let a function f ∈ A satisfy
0
zf (z)
z 2 f 00 (z)
1
α
<
+α
> αλ λ −
+λ− .
f (z)
f (z)
2
2
Then f ∈ S ∗ (λ).
Setting β = 0 and λ = α/2, 0 < α < 2 in Theorem 2, we obtain the following
result of Li and Owa [3].
Corollary 5 If a function f ∈ A satisfies
0
zf (z)
α2
z 2 f 00 (z)
<
> − (1 − α),
+α
f (z)
f (z)
4
0 < α < 2,
then f ∈ S ∗ (α/2).
Setting β = α = 1 in Theorem 2, we obtain the following result.
Corollary 6 For all z ∈ E, let f in A satisfy the condition
(
λ
> λ − 2(1−λ)
, 0 ≤ λ < 1/2,
zf 00 (z)
< 1+ 0
(1−λ)
f (z)
> λ − 2λ , 1/2 ≤ λ ≤ 1.
Then f ∈ S ∗ (λ).
References
[1] S. Fukui, On α-convex functions of order β, Internat. J. Math. & Math.
Sci., 20, 4, 1997, 769-772.
[2] Z. Lewandowski, S.S. Miller, E. Zlotkiewicz, Generating functions for
some classes of univalent functions, Proc. Amer. Math. Soc., 56, 1976,
111-117.
[3] J.-L. Li, S. Owa, Sufficient conditions for starlikeness, Indian J. Pure
Appl. Math., 33, 2002, 313-318.
[4] S. S. Miller, P. T. Mocanu, Differential subordinations and inequalities
in the complex plane, J. Diff. Eqns., 67, 1987, 199-211.
58
S. Singh, S. Gupta, S. Singh
[5] S. S. Miller, P. T. Mocanu, M. O. Reade, All α- convex functions are
univalent and starlike, Proc. Amer. Math. Soc., 37, 1973, 553-554.
[6] S. S. Miller, P. T. Mocanu, M. O. Reade, Bazilevic functions and generalized convexity, Rev. Roumaine Math. Pures Appl., 19, 1974, 213-224.
[7] P. T. Mocanu, Alpha-convex integral operators and strongly starlike functions, Studia Univ. Babes-Bolyai Math., 34,2, 1989, 18-24.
[8] K. S. Padmanabhan, On sufficient conditions for starlikeness, Indian
J.Pure appl. Math., 32,4, 2001, 543-550.
[9] V. Ravichandran, C. Selvaraj, R. Rajalakshmi, Sufficient conditions for
starlike functions of order α, J. Inequal. Pure and Appl. Math., 3(5)
Art. 81, 2002, 1-6.
Sukhwinder Singh
Department of Applied Sciences
Baba Banda Singh Bahadur Engineering College
Fatehgarh Sahib -140407 (Punjab), India
e-mail: ss billing@yahoo.co.in
Sushma Gupta, Sukhjit Singh
Department of Mathematics
Sant Longowal Institute of Engineering & Technology
Longowal-148106 (Punjab), India
e-mail: sushmagupta1@yahoo.com, sukhjit d@yahoo.com