http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
""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.""--Provided by publisher.
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}.
Chapman & Hall/CRC Pure and Applied Mathematics