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Abstract
Let H(U)
be the space of analytic functions in the unit disk
U and let h E H(U).
We define a subset
K({J,-r),h C H(U)
such that the
operator
A({J,"Y),h : K({J,"Y),h --+ H(U)
given by
A({J,"Y),hU)(Z) = [~"Y~Z~ faz
jf3(t)h-r-1(t)h'(t)dt]
1/f3
is well defined. Then we determine
a class of functions whose images by
A({J,-r),h operators
are univalent.
In addition,
we give some particular
cases of our main result obtained
for appropriate
choices of h, {3and
"y.
Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 3.
Lemma 3. [10, p. 159 The function L(z;t) = al(t)z +... with al(t) =J 0 for all t ~ 0 and Hm lal(t)1 = +00 is a subordination chain if and…
Lemma 3. [10, p. 159} The function L(z;t) = al(t)z + ... with al(t) =J 0 for all t ~ 0 and Hm lal(t)1 = +00 is a subordination chain if and only if t--t+oo [ 8L/8z] Re
Lemma 4.
Lemma 4. [11, Corollary 3 Let Re 3 > -"2 and for F E H(U), with F' (0) =J 0 let zF'(z) zFI/(z) (2.3) J( 3, F)(z) = ( 3 - 1) F(z)
Lemma 4. [11, Corollary 3} Let Re {3 > -"2 and for F E H(U), with F' (0) =J 0 let zF'(z) zFI/(z) (2.3) J({3, F)(z) = ({3 - 1) F(z)
Lemma 5.
Lemma 5. [5, Theorem I Let 3" E C with 3 =J 0 and let h(z) = c+h1z+·.. be analytic in U. If Re [ 3h(z) +,] > 0, z E U then the solution of…
Lemma 5. [5, Theorem I} Let {3" E C with {3 =J 0 and let h(z) = c+h1z+· .. be analytic in U. If Re [{3h(z) +,] > 0, z E U then the solution of the differential equation zq'(z) .
Theorem 1.
Theorem 1. Let /3" E C with Re (/3 +,) > 0 and let h E A. Then the integral operator given by (1.1) is well defined on the subset…
Theorem 1. Let /3" E C with Re (/3 + ,) > 0 and let h E A. Then the integral operator given by (1.1) is well defined on the subset K(f3,-y),h= {f E H(U): f(O) = 0,1'(0) -I O,/3z;~~~) + J(r,h)(z) --< Rj3+-y(Z)}.
Theorem 2.
Theorem 2. Let /3" E C with /3 +, > O. For a function h E A we denote by m=inf Re,z:~~~):ZEU andby M=sUP Re,z:~~~):ZEU. Let 8 be a real…
Theorem 2. Let /3" E C with /3 +, > O. For a function h E A we denote by m=inf{Re,z:~~~) :ZEU} andby M=sUP{Re,z:~~~) :ZEU}. Let 8 be a real number such that (3.1) m - (/3+,) < 8 ~ min { m; m _ /3+; - 1 } and
Corollary 1.
Corollary 1. Let f3, 'Y E R with f3 + 'Y ~ 1 and let A E C with IAI::; 1 such that and 2hllAI2 - (f3 - 'Y+ 2hl + 3)IAI+ f3 - 'Y+ 1 ~ O. If…
Corollary 1. Let f3, 'Y E R with f3 + 'Y ~ 1 and let A E C with IAI ::; 1 such that and 2hllAI2 - (f3 - 'Y+ 2hl + 3)IAI+ f3 - 'Y+ 1 ~ O. If f E H(U) with f(O) = 0, f'(O) -# 0 and zj'(z) 1 .
Corollary 2.
Corollary 2. Let 3,,' E R with'Y 2 0 and 3+ 'Y2 1 and let>. E C such that -'Y - 1 1 - 1>'1 1 3+ ~( ~ + 1+ 1>'1 2 -2" - -2- and If j E H(U)…
Corollary 2. Let {3,,' E R with'Y 2 0 and {3+ 'Y2 1 and let>. E C such that -'Y - 1 1 - 1>'1 1 {3+ ~( ~ + 1+ 1>'1 2 -2" - -2- and If j E H(U) with f(O) satisfies
Function classes studied:
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