Differential Subordination's : Theory and Applications /

""Examining a topic that has been the subject of more than 300 articles since it was first conceived nearly 20 years ago, this monograph describes for the first time in one volume the basic theory and multitude of applications in the study of differential subordination's.""-...

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Bibliographic Details
Main Authors:	Miller, Sanford S. (Author), Mocanu, Petru T. (Author)
Corporate Author:	Taylor & Francis
Format:	eBook
Language:	English
Published:	Boca Raton, FL : CRC Press, 2000.
Edition:	First edition.
Series:	Chapman & Hall/CRC Pure and Applied Mathematics
Subjects:	Numerical analysis. Computational Numerical Analysis Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

chapter 1 PRELIMINARIES
1.1. A Short History
chapter r(c) ta-1(t)c-a-1
~ r(a + r(b +
chapter 2 THEORY OF SECOND-ORDER DIFFERENTIAL SUBORDINATIONS
2.1. Introduction
chapter + anzn + ··· be analytic in U with
chapter q'(,
chapter + anzn + · · ·
If not u au
chapter dU that p(zo)
lzl < lzlp(z =I = M,
chapter 2 3. Admissible Functions and Fundamental Theorems
chapter If p .JI[a, and 'lf(p(z), zp'(z), z
chapter = e I
n, = z U and
chapter 2 4. Examples
J 1}, by using (2.2-7) and (2.2-8) the condition of admissibility z and n
chapter +vi; z)] = a+
= Re[pi+a(z)·afpi] =
chapter l 'lf(pi,a,JL+vi;z)l;;?: 1, -n[ 1 + p ]/2,
I + Szp'(z)-
chapter fi[O, 1], then I zp
z in a )] ), ]/2. The second condition in (2.4-7) implies that If p 1 , n]
chapter 2 5. The Open Door Lemma and the Integral Existence Theorem
]
chapter f(z)
= 2enz/(1- = 2e If 0, = b + = + = 0, lim =. This leads to and lim
chapter I I < + 2cfn = en
chapter + 8), n]. By differentiating (2.5-18) we
= a.+ =
chapter .A, then the following sharp implications hold:
= ( ( ) '(
chapter 3 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL SUBORDINATIONS
3.1. First-Order Linear Differential Subordinations
chapter of Theorem 2.3d. We only need to show that
I arg [m·P(z)] I < 2·
chapter Proof. 'V(r,s)
= + sfy, = + ( ml n 1, we conclude that
chapter if y 1, then this theorem reduces to Robinson's differential sub-
z ]II ]a and q(z) 1-z 1-z
chapter If p(z0, then ±1. If we let ri
chapter = = = .A and
= + y] I [ zH'(
chapter If we apply this last theorem to the special relationship between the integral = f]
of Theorem 3.2j, with f] .A = zK'
chapter 3 3. Briot-Bouquet Applications in Univalent
_1 I / J[ J
chapter z)t-
chapter = + = + Bz'
-1/2 and If of Lis the Libera integral operator defined by
chapter = [f], f *. These functions F and
f satisfy Bz)[l-
chapter fort 0, and lim aI
z and t 0
chapter + y) -t corresponds to the Briot-Bouquet
If zU, = 0.
chapter + + 9'[q(z)]
If =
chapter 0 Combining all of these cases we have the following result:
if (1/2, lfp p(z) 0,
chapter < a1/3. Therefore, by Theorem 3.4i we deduce the
chapter If f S AI 2 -
chapter If = =
f](z) [ (a +
chapter *. The
f to be in the set of convex functions and determine
chapter a, y, o and a be real numbers satisfying
s*,g l[f,g] S*. In addition, if l[f,g] = (g(z)/ O+aRe[zg'(z)_1] o-a.
chapter = if *,
chapter If = = o = = =
chapter fCJ(
If = J(
chapter 74 ] obtained specific
3.6 Subordination-Preserving Integral Operators
chapter It follows immediately from Lemma 1.2c that the integral operator I as given
chapter + p(z) + -
chapter 4 APPLICATIONS OF SECOND-ORDER DIFFERENTIAL SUBORDINATIONS
I 1+ I y 1+ 2 Im · Im y y I 2Re y, I + I P - y I )
chapter = + z)/(1-
chapter + B(z)zp'(z) + C(z)p(z) + D(z) Mz
chapter I + C(z)p(z) + D(z) I < M,
chapter (U) at
If W, Z C and -1 = -1)] I < rt/2. Using
chapter = At + +
z/(B + A simple calculation shows that
chapter e with s; 1 satisfies (4 .2-3), we obtain:
if Tis = I < 1/4, If =
chapter If = =
chapter 4 3 Integral Operators Preserving Bounded Functions
chapter P= =
I I < N.
chapter f(J),
chapter if y2:1/2 and r
chapter = = -
A. If
chapter p + y + + +
chapter a-[ + /2, a + 0 and z U. Hence, by Theorem 2.3i
chapter If c 1 + N(a -
chapter If cR(a), where R(a) is given by (4.5-10), then
Jt-l/2(1- 5/4. In the special case c = 3/2, iz
chapter < b < c and a [-2, 0). By employing (1.2-14) and
z U. Using this result in the above equation we obtain F'(a,b,c;z) 0 for z U.
chapter 4 6. The Schwarzian and Starlikeness
chapter 5 SPECIAL DIFFERENTIAL SUBORDINATIONS
5.1. Conditions for Special Subclasses of Starlike Functions
chapter If = ( + az ), then
+ naf2, if < If a
chapter If =
in that * [ ]
chapter If f
S*[p].
chapter 14 ] and in 1991 [ 228] the result M
chapter .fi c; z) is the confluent hypergeometric function given in (1.2-3).
chapter f
1/
chapter an n /ln 4 when an is given by (5.2-15) and
chapter 5 If A2
zj
chapter a(k).
)ln[l-l/k] = if < [I+ f S*( If .A, j(z)· +II < I zj'(z) 0, when a -
chapter 5 5. Functions with Bounded Turning And Starlike Functions
I arg f'(z) I rt/2, and arg f'(z) is the angle of turn (or rotation) of the z under the mapping f . It is well known that f S *. This section describes some of
chapter + z ]
I + arg (p(z) zp'(z)) I I arg p(z) I I arg I f .S *.
chapter f e .Aand
= < -
chapter Wim If
I I < IP(z)- I I < N
chapter = = If
chapter 1 <
FE S*. If y) if
chapter 1 e .A satisfy the differential subordination (5.6-6). Using the
= = and = F(z) f'(z) = f(z).
chapter f; z)
chapter his convex it is easy to show that (5.6-15) implies
chapter 6 HIGHER ORDER DIFFERENTIAL SUBORDINATIONS
6.1. Introduction
chapter )e-i
-1),
chapter f is analytic and has a zero of order k, then
= + + f J./ and 1
chapter ].If {d r lr o is a convex null sequence,
chapter 7 If B
7.1. Preliminary Lemmas
chapter F: e and let 0 be a complete
)/dz] 0, and I F(wz I 1, for w e U. Since F(O) = [I lim(l-r)-
chapter S au
f( P.n) q(V), then there exist points , au f(zo)
chapter = If z
II}, ),/(zo)), I : I ' I I 'o I } I 'o ) I .
chapter en with the supremum norm
f(zo) = {II f(z) : },
chapter < r< 1, zJBr. and let
= {II /(z) II: II}, /IIJ(z = mj(z
chapter f is locally biholomorphic at z
zpoint of maximum the = mf(zo), = 1/ Df(z A 0.
chapter I ' I z
u '0 {I g(') I:
chapter < M, and let 'I'[M]. If the differential equation
llf(z) + f(z)(z f(z) M.
chapter 7 .3. Dominants and Admissible Functions in
f and g be f is said to be subordinate tog, written f g or /(z) g(z), if there exists a mapping ..J./(B), with = f(B) f g if and only if f(O) [50]
chapter fi(B) and biholomorphic
chapter c en and let g be a biholomorphic
chapter = { (u, en en: II= and vII }·
[Dg(')r II' = Iff =
chapter f Then there is ae2, · · · , n} such that
C·zf' h(z),
chapter 8 APPLICATIONS
SUBORDINATIONS 8.1. Harmonic Functions
chapter If we let p(z) = v + iv(z), then p will be analytic in U, p(O) = 1,
)J/2,
chapter + 2(Im P(z ))p + 1 -
f e 1:and f'(z) +
chapter THEOREM 8.2j. [ 247 ] Let a < 1, 0 < y, f and
= + y) = }:.r f then
chapter 9 ] [ 254] e then
f, g e JJ
chapter 'I': CxU ~ C that
+kim + Cim + Im 0, = = 0,
chapter If a1/2, 0 and
= 0, a(llf(z)ll + llzf'(z)ll) + Plzl < f(z)ll < I. = 1/2 =
chapter APPENDIX CONVEXITY OF BERNOULLI FUNCTIONS
= z/(ez - lzl < 2n, and in such a 2 + z '
chapter f is convex, but that f is convex f is not known.
f in I (e-z+z-1)z
chapter BIBLIOGRAPHY
chapter a, L'A nalyse Numerique et la Theorie de
chapter ala theorie des foncions univalentes, Casopis
chapter LIST OF SYMBOLS
=..A, A[f]
chapter L [f]
Ly[f] L(z,t)
chapter 452 S[il]
S*[p] s*n.An = S*(a)n.An s*[il] U=U {o}.