Starlike and convex maps in Banach spaces

LEMMA 1. "Let v(x, t): B x I—+B be holomorphic for each te I = [0,1], v(091) = 0 αwd v($, 0) = x. If limt^+ [(x — v(x9 is holomorphic in B then w e$

LEMMA 2 Let f: B—>Y be a biholomorphic map (holomorphic with a holomorphic inverse) of B onto an open set f{B) c Y and let /(0) = 0. Let F{x, t): B x I —+Y be a holomorphic function of x for each t e I, F(x, 0) = f(x), F(0, ί) = 0 and suppose F(B, t) c f{B) for each te I. Further, suppose limt_>0+[(F(xt 0) — F(x, t))/t] = F(x) exists and is holomorphic. Then F(x) — Df(x)(w(x)) where w e ^0(B) (Df(x) is the Frechet derivative of f at x).

LEMMA 3. If we&*0(B) and\a\ < 1 then — w ( a x ) e ^ 0 ( B ) (—w(ax) is understood to be the limit value Dw(O)(x) when a = 0) . Further, if x' e T(x), 0 < ||a?|| < 1 then x r{w{x)) = 0 if and only if x'(Dw(O)(x)) = 0 and in this case x'(l/a-w(aχ)) = 0 when \a\ < l/||aj||.

THEOREM 1. Suppose f:B—*Y is starlike and that /~ x is holomorphic on f(B) open c Γ , There exists we^(B) such that (1) f(x) = Df(x)(w(x)).

THEOREM 2. Let f:B~>Y be holomorphίc and /(0) = 0. Assume Df(x) has a bounded inverse for each xeB (hence f" 1 exists and is holomorphίc in a neighborhood of each point of f{B)) and that for some w G ^(B), f(x) = Df(x)(w(x)). Suppose further that for each r, 0 < r < 1, there exists M(r) such that \\[Df(x)]~ l\\ ^ M(r) when \\x\\ <L r. Then f is starlike.

LEMMA 4. If f: B—*Y is a biholomorphic map of B onto a convex domain then f(Br) is convex for each r, 0 ^ r ^ 1.

THEOREM 3. If f: B—>Y is convex then ( 3) D*f{x){x, x) + Df(x)(x) = Df(x)(w(x)) where w e

LEMMA 5. If we QQ(B) and \ a\ < 1 then l/a w(ax, ay) e Q0(B) (the

THEOREM 4. Suppose f:B-+Y is convex. Then f(x) - f(y) = Df(x)(w(x, y)) where w e Q(B).

THEOREM 5. Suppose f: B-+Yis holomorphic, Df{x) has bounded inverse for each xeB and that for some w eQ(B), f(x) — f(y) = Df(x)(w(x, y)). Suppose further that for each r, 0 < r < 1, there exists M(r)>0 such that ||[A/WΓΊ! ^ M(r) when | | α j | | ^ r . Then f is convex.

THEOREM 6. Let f: B-+Y be convex, 0 Φ X e B and let x' e T(x). Then the hyperplane {y e Y: x f[(Df(x)y i(y)] — ^ r[(Df(^)y i(f(x))]} is a supporting hyperplane for the convex set f(B{]x]{). If y Φ 0, x'(y) — 0 and \\x + ty\\ = ||α;|| for 0<t< t0, then x f[(Df(x))' 1(f(x + ty))] = x

THEOREM 7. Let X = L(μ) (the space of complex valued integrable

COROLLARY 1. If X = s ι and f: B~*Yis convex then f(x) — /(0) is linear.

THEOREM 8. Let X — L°°{μ) {the space of essentially bounded, complex valued, measurable functions on S) and suppose f: B—>Y is convex. If x,yeB where x(s) = y(s) for all seEaS and μ{E) > 0 then {Dfφ))- ι{f{x)){s) = φ / i O))" 1/^/)^) for almost all seE.

COROLLARY 2. IfX= s~ and f:B—*Y is convex then f(x) - /(0) = Df(0)(g(x)) where g(x) = {g^x,), g2(x2), •••) and gk(xk) = xk + + maps I % I < 1 (mίo a convex domain. 4. Discussion. The techniques used in this paper may also be used to extend the concepts of close-to-convexity and spirallikeness to Banach spaces. There are at least two different ways of extending the analytic condition for close-to-convexity to Banach spaces and these