# Distortion Theorems in Relation to Linear Integral Operators by Yûsaku Komatu (auth.)

By Yûsaku Komatu (auth.)

The current monograph involves elements. ahead of half I, a bankruptcy of advent is supplemented, the place an outline of the total quantity is given for reader's comfort. the previous half is dedicated frequently to show linear inte gral operators brought by means of the writer. numerous homes of the operators are confirmed, and specializations in addition to generalizations are tried variously with a purpose to make use them within the latter half. in comparison with the previous half, the latter half is de voted almost always to increase numerous sorts of distortions below activities of fundamental operators for varied frequent functionality additionally absolute modulus. genuine half. diversity. size and quarter. an gular spinoff, and so forth. in addition to them, distortions at the category of univalent capabilities and its subclasses, Caratheodory category in addition to distortions through a differential operator are handled. similar differential operators play additionally lively roles. Many illustrative examples might be inserted for you to support realizing of the overall statements. the elemental fabrics during this monograph are taken from a sequence of researches played through the writer himself mainly long ago twenty years. whereas the subjects of the papers pub lished hitherto are inevitably no longer prepared chronologically Preface viii and systematically, the writer makes right here an attempt to ar variety them as ,orderly as attainable. In attaching the import ance of the self-containedness to the publication, a few of unfamil iar topics can also be inserted and, additionally, be absolutely followed by means of their respective proofs, notwithstanding unrelated they could be.

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Proof. {-1]1 t 0 log- U - 1) 2 t a - 1 ( log - 2 + Here we remember f E f (a - l) t a - and a ) o. {-l ) dt . { 1 Chapter 2. (a)Af(z) U - - f 1) f (zt ) t a - I (a - f 1) I f (zt ) t 2( log - a - 2 1 )A-2 t (log - 1 t at ) A-I dt ) o In the following lines, we shall. £ ( a) in Lerms of J defined by we attempL to derive an expression for and i LS iterations. § It is 45 7. • ). in particular, 1 O. £ ( a) and THEOREM 7. e have Chapter 2. properties of integral operators 46 ... lar, ~'hel1 a) /C- 1 J z/C x;=l /C.

P( Proof. Ie £(k). Ie = (k ) 1, 2, ... ain CD £ [pJ I (f) k=I "" I Je=1 (f}1 £(k) k (Je zJe 1 k =2 fiJ "" I Je=I JJe k -1 "" + 1) ! zJe k k r Je=I (- I)Je-I (k (k - - Je - 2) ! 1) ! z Je ,JJe Chapter 2. Properties of integral operators 54 where the coefficients of the last expression are given by (JC - 1)! (01 + (-1) JC-1 k I=JC+1 (k - 2)! k--------------- (0 (k-JC-I)! k t =1 I k =2 We thus get the desired result. I. her generalized with respect to the referring measure. 2. Let a probab1Lity measure a(t) where a measure = f'o" t a 0" be g1ven by dda) def1ned on the 1ntervaJ r (0, "") the cond1 t1ons pet) =O"'(t J~ o a t 0" (1) a-I f»o drCa) dr(a) ~ (t E I), 0 1.

He first. he Theorem. , we have § If. ( a r For any e - f CO vI > 0 5. Operator generated by (zt ) t I[Af(z)