Far East Journal of Mathematical Sciences (FJMS) Volume 76, Number 1, 2013, Pages 199-204 Published Online: June 2013 Available online at http://pphmj.com/journals/fjms.htm Published by Pushpa Publishing House, Allahabad, INDIA
STARLIKENESS OF CLASSES OF FUNCTIONS DEFINED BY CONVOLUTION AND SUBORDINATION Samaneh G. Hamidi1, Suzeini A. Halim1 and Jay M. Jahangiri2,* 1
Institute of Mathematical Sciences Faculty of Science University of Malaya 50603 Kuala Lumpur Malaysia e-mail:
[email protected] [email protected]
2
Mathematical Sciences Kent State University Burton, Ohio 44021 U. S. A. e-mail:
[email protected] Abstract Certain classes of functions defined by convolution and subordination are considered. Necessary and sufficient containment conditions for these classes of functions are defined which in turn will lead to the starlikeness of such functions.
© 2013 Pushpa Publishing House 2010 Mathematics Subject Classification: 30C45, 30C50. Keywords and phrases: subordination, Hadamard product, starlike functions. *Corresponding author Submitted by K. K. Azad Received October 9, 2012
Samaneh G. Hamidi, Suzeini A. Halim and Jay M. Jahangiri
200
1. Introduction Let A denote the class of functions that are analytic in the open unit disk
U = {z : z ∈ C, z < 1} and A0 = { f ∈ A : f (0) = f ′(0) − 1 = 0, z ∈ U }. For f and F in A, we say that f ( z ) is subordinate to F ( z ) , written f ≺ F , if there exists a Schwarz function w( z ) , w(0) = 0 and w( z ) < 1 in U such that f ( z ) = F ( w( z )). The Hadamard product or convolution of two power series
∞
f ( z ) = z + ∑n = 2 an z n and F ( z ) = z +
f (z) ∗ F (z) = ( f ∗ F ) (z) = z +
∞
∑n = 2 bn z n
is given by
∞
∑n = 2 anbn z n .
In the sequel, unless otherwise stated, g will denote a fixed function in A0 and h will always denote a convex univalent function in U with h(0) = 1 and Re h( z ) > 0, z ∈ U . For f ∈ A0 such that ( g ∗ f ) z ≠ 0 and ( g ∗ f )′ ≠ 0 in U, the following classes S ( g , h ) , K ( g , h ) and J α ( g , h ) are defined by Shanmugam [2]:
⎧⎪ ⎫⎪ z ( g ∗ f )′ ( z ) S ( g , h ) = ⎨ f ∈ A0 : ≺ h( z )⎬ , (g ∗ f ) (z) ⎪⎩ ⎪⎭ ⎧⎪ ⎫⎪ z ( g ∗ f )″ ( z ) K ( g , h ) = ⎨ f ∈ A0 : 1 + ≺ h( z )⎬ , ⎪⎩ ⎪⎭ ( g ∗ f )′ ( z ) ⎧⎪ z ( g ∗ f )′ ( z ) J α ( g , h ) = ⎨ f ∈ A0 : (1 − α ) (g ∗ f ) (z) ⎪⎩ ⎛ ⎫⎪ z ( g ∗ f )″ ( z ) ⎞⎟ ( ) , α ∈ + α⎜⎜1 + ≺ h z R ⎬. ⎪⎭ ( g ∗ f )′ ( z ) ⎟⎠ ⎝ Note that J 0 ( g , h ) ≡ S ( g , h ) , J1( g , h ) ≡ K ( g , h ) and J α ( g , h ) provide a continuous passage from the class K ( g , h ) to S ( g , h ) as α decreases from
Starlikeness of Classes of Functions Defined by Convolution …
201
1 to 0. Recall that a function f ∈ A0 is starlike univalent ( S ∗ ) if and only if Re ( zf ′( z ) f ( z )) > 0 and convex univalent ( K ) if and only if
Re (1 + zf ′′( z ) f ′( z )) > 0, z ∈ U . In this paper, we introduce necessary and sufficient conditions for functions in J α ( g , h ) and examine the impact of these conditions on the starlikeness of classes of functions defined by convolution and subordination. 2. Main Results
To prove our theorem, we shall need the following two lemmas, the first of which is due to Miller and Mocanu [1]. Lemma 2.1. Let α and δ be so that α + δ > 0 and let the function φ be
analytic in U so that φ(0) = 1 and φ( z ) ≠ 0 in U. For f ∈ A0 , let α
zf ′( z ) zφ′( z ) + + δ ≺ Qα + δ ( z ) f (z) φ( z )
(1)
and define F by 1 (α + δ ) z ⎛ ⎞ F ( z ) = ⎜ (α + δ ) f α (t ) t δ −1φ(t ) dt ⎟ . 0 ⎝ ⎠
∫
Then F ∈ A0 , Re((α + δ ) zF ′( z ) F ( z )) > 0 in U, and F ∈ S ∗ . The “open door function” Qc (see Miller and Mocanu [1]) is defined by z+b⎤ Qc = Qc ( z ) = H ⎡ , ⎢⎣1 + b z ⎥⎦
z ∈ U,
where H ( z ) = 2 Nz (1 − z 2 ) , N = N (c ) = [ c 2 1 + 2 Re c + Im c ] Re c, c is a complex number such that Re (c ) > 0, and b = H −1(c ).
The condition (1) involving the slit mapping Qα + δ opens up the left-half plane through the “open door” between the two slits of Qα + δ .
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Samaneh G. Hamidi, Suzeini A. Halim and Jay M. Jahangiri Lemma 2.2. For α > 0, f ∈ J α ( g , h ) if and only if k ∈ S ( g , h ) , where α
1 ⎞ ⎛1 z ( g ∗ k ) α dt ⎟⎟ . ⎝α 0 ⎠
( g ∗ f ) ( z ) = ⎜⎜
∫
Proof. Suppose k ∈ S ( g , h ). Then (2) implies
′ 1 ⎛ ⎞ 1 α (g ∗ k ) (z) ⎜⎜ ( g ∗ f ) α ( z ) ⎟⎟ = z α ⎝ ⎠ which gives α (g
1− α ∗ k ) ( z ) = z ( g ∗ f )′ ( z ) ( g ∗ f ) α ( z ).
We can then write
( g ∗ k ) ( z ) = z α (( g ∗ f )′ )α ( z ) ( g ∗ f )1− α ( z ) α ⎛ z ( g ∗ f )′ ( z ) ⎞ ⎟ . = (g ∗ f ) (z) ⎜ ⎜ (g ∗ f ) (z) ⎟ ⎝ ⎠
Thus, we have
( g ∗ k )′ ( z ) α ⎛ z ( g ∗ f )′ ( z ) ⎞ ′ ⎟ = (g ∗ f ) (z) ⎜ ⎜ (g ∗ f ) (z) ⎟ ⎝ ⎠
⎛ (( g ∗ f )′ ( z ) + z ( g ∗ f )″ ( z )) ( g ∗ f ) ( z ) − z (( g ∗ f )′ ( z ))2 ⎞ ⎟ + α⎜ ⎜ ⎟ g ∗ f ) (z) ( ⎝ ⎠ α −1 ⎛ z ( g ∗ f )′ ( z ) ⎞ ⎟ . ⋅⎜ ⎜ (g ∗ f ) (z) ⎟ ⎝ ⎠
Dividing ( g ∗ k )′ ( z ) by ( g ∗ k ) ( z ) , we obtain
( g ∗ k )′ ( z ) ( g ∗ f )′ ( z ) α α( g ∗ f )″ ( z ) α( g ∗ f )′ ( z ) = + + − . (g ∗ k ) (z) (g ∗ f ) (z) z (g ∗ f ) (z) ( g ∗ f )′ ( z )
(2)
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203
Hence ⎛ z ( g ∗ k )′ ( z ) z ( g ∗ f )′ ( z ) z ( g ∗ f )″ ( z ) ⎞⎟ = (1 − α ) + α⎜⎜1 + . (g ∗ k ) (z) (g ∗ f ) (z) ( g ∗ f )′ ( z ) ⎟⎠ ⎝
(3)
The right hand side of equation (3) is subordinate to h( z ) in U, since its left hand side is according to our assumption. Therefore, f ∈ J α ( g , h ). Next, suppose that f ∈ J α ( g , h ). Then, by definition,
(1 − α )
⎛ z ( g ∗ f )′ ( z ) z ( g ∗ f )″ ( z ) ⎞⎟ + α⎜⎜1 + ≺ h ( z ). (g ∗ f ) (z) ( g ∗ f )′ ( z ) ⎟⎠ ⎝
Using a reverse argument justifies the “Only if” part of the theorem. Now, we are ready to state and prove our starlikeness theorem. h( z ) ≺ αQ 1 ( z ) and
Theorem 2.3. If α > 0,
α
f ∈ J α ( g , h ) , then
∗
(g ∗ f ) ∈ S . Proof. Let f ∈ J α ( g , h ) , α > 0. Then, by Lemma 2.2, there exists a
function k ∈ S ( g , h ) , where α
1 ⎛ 1 z −1 ⎞ t ( g ∗ k ) α (t ) dt ⎟⎟ . ⎝α 0 ⎠
∫
( g ∗ f ) ( z ) = ⎜⎜
Upon differentiation and using a simple algebra, we obtain
z ( g ∗ f )′ ( z ) ≺ h ( z ) ≺ α Q 1 ( z ). (g ∗ f ) (z) α
Finally, by applying Lemma 2.1, we conclude that α
1 ⎛1 z ⎞ ( g ∗ f ) ( z ) = ⎜⎜ ( g ∗ k ) α (t ) t −1dt ⎟⎟ ∈ S ∗. α 0 ⎝ ⎠
∫
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Samaneh G. Hamidi, Suzeini A. Halim and Jay M. Jahangiri References
[1] S. S. Miller and P. T. Mocanu, Classes of univalent integral operators, J. Math. Anal. Appl. 157 (1991), 147-165. [2] T. N. Shanmugam, Convolution and differential subordination, Internat. J. Math. Math. Sci. 12(2) (1989), 333-340.
