Sharp starlikeness conditions for analytic functions with bounded derivative

THEOREM 1.1. Letbe&0 and :=sup f \b(tz)\dt, zeD Jo fi := 1/Vl + c2. Then c < 1/2 and ^(b) C &*, where &* is the set of starlike univalent functions in £/. If (1.2) b{t) = max l*(reI>)|, 0 < t < 1, then the constant fi cannot be replaced by any larger number without violating the conclusion. REMARKS. (1) Various choices of b lead to function classes which are commonly investigated. For instance the choices b(z) = z" restrict the functions under consider- ation to those with

Theorem 1.1, which we do not include, is to get bounds for starlikeness of order a (Re zf'(z)/f (z) > a). For the case b(z) = z this was already done by Fournier [2]. We begin with the study of the sets : f and to this end we also define 1 + Xcw : z, w € Note that S2x,c» as the union of circular discs (with w as parameter) containing the point 1, is a domain, starlike with respect to 1.

THEOREM 1.2. (i) LetO^be 3SQ and c := supzeD /„' \b(tz) | dt. Then rx(b) c fiiiC, 0 < A. < 1. (ii) Furthermore, for each f e ^(b) we have (iii) If, in addition, b satisfies the condition in (1.2), then (1.5) r x ( b ) =-fiI7> 0 < k < l . The condition (1.2) holds for b(z) — z, so that (1.5) applies to <^. In Theorem 2.1 below we shall describe the boundary of QKc in parametric form, and this makes it https://doi.org/10.1017/S1446788700002482 Published online by Cambridge University Press

COROLLARY 1.3. Letb€@o and c be as before. For 0 < a < 1 let sin (an /2) fi(oc) := y/l + 2c cos (an/2) + c2' Then f € ^a)(b) implies that f is strongly starlike of order a. Ifb satisfies (1.2), then the constant n(a) cannot be replaced by any larger number without violating the conclusion, although every single function f e ^(^(b) is strongly starlike of order af < a. The result in Corollary 1.3, without the sharpness part, was previously obtained by Ponnusamy and Singh [5]. A function g e &f is

COROLLARY 1.4. Let b € @0 and c be as before. Iff € &i/(c+2)(b) andf = zg' then g is uniformly convex. Ifb satisfies (1.2), then the constant l/(c + 2) cannot be increased without violating the conclusion. Taking b(z) = z" in Corollary 1.4 we find that g e stf is uniformly convex if / = zg' satisfies and the bound is best possible. Non-sharp results for this case can also be found in [5]. A different situation where our method can be applied as well is the case of uniformly starlike functions /

THEOREM 1.5. Let b € &0 and \x starlike, and, for every single b € property. However, if with b e 39 1/V2. Then each f € ^(b) is uniformly the number fj, is the largest one with this Ht)w(t) dt e satisfies / ' € C°(lD>), then there exists /if > /x such that z + (1/ J* b(t)w(t) dt is uniformly starlike as well. This result is somewhat surprising, as it implies that, for instance, a restriction to functions of the form (1.3), however large n may be, does not increase the optimal value of (i. One o

Theorem 1.5 (with b(z) = z) and that his observation reflects a more general fact. Our final result deals with functions / € &/ starlike with respect to symmetrical points, which are defined by the condition 2zf'(z) Re fiz)-f(-z) > 0, z € D. https://doi.org/10.1017/S1446788700002482 Published online by Cambridge University Press

THEOREM 1.6. Letbe^0andc:=(l/2)supzeDf^ \b{tz)\dt, ix(c):= 1/Vl+c2. Then f e ^^(b) implies that f is starlike with respect to symmetrical points. Ifb is an even function satisfying (1.2), then the constants fi(c) are best possible. This result says, for instance, that/ 6 ^./-/ils starlike with respect to symmetrical points, but this is not known to be best possible (b(z) = z is not even). However, if / € <^/Vio is °f m e form / (z) = z+a3z3+a*z4-\ , then / is starlike with respect to symmetrical

THEOREM 1.7. Letx, y e D, Z\, Zi e 3D andb e 88. Then there exists a sequence of functions Vk e 3& n J f (5) such that Vk(l) = z, and (1.7) lim r b{t) Vk(t)dt = zi T b(t)dt. k ° J J = zi T Jx The functions Vk can be chosen independently ofx, y. The proof of Theorem 1.7 will be given in the Appendix. 2. Proof of Theorem 1.2 and its corollaries PROOF OF THEOREM 1.2. (i)Fixa € r\(6), so that there exists/ e &i(b),zo e D, with a = Zof'(zo)/f(zo)- We write b(t)w(t) dt = z + kF(z). Jo

THEOREM 2.1. For 0 < c < 1 and 0 < A. < 1, cA. ^ 1, the boundary ofQ,x,c is described by (x(0), y(9)), 9 € [0, 2n), where l+kcos9 -^k2 sin2 9 + ckcos9jl +k2+2kcos9 -dk2 sin2 9 (2.4) x(e)= r^x2 ' y(0) = (*(<?)-1) tan 6>. PROOF. Let G(9, <p) - (1 + kew)/(l + ckei<p). For fixed 9 the function G(9, <p) describes the circle (2 5) Ix 1 I l y - - \2_c2k2(l+k2

THEOREM 2.2. (i) For c> 0 and 0 < a < 1 let sin (an/2) (2.7) k(a) := +2ccos(an/2) Then \ arg co\ < an/2 holds for every u> € (ii) For zo = ei<p", w0 = e^0, with <p0 = sin"1 (VI - k(a)2) e (n/2,n) and Q = (pQ — (1 + a/2)n, we have arg- l+xk(a)zo +xck(a)w0 n > a-, x > 1.

Theorem 1.5 follows readily from (3.3). • PROOF OF THEOREM 1.6. Let f Jo Then, by Theorem 2.2 with a = 1, 2zf'(z) _ Ke = Ke b(t)w(t)dt e Z 1 + n(c)b(z)w(z) = K e

Theorem 1.7 with x = — 1, y = 1 produces a sequence of functions Vk e 3S such that for every x > 1 l+xn(.c)Kz)Vk(z) lim Re < 0 for z e proofs. {x^c)/2)j\b{tz)Vk{tz)dt »close enough to 1. We omit the details which are similar to those in previous • 4. Appendix PROOF OF THEOREM 1.7. We may assume that z\ = 1. If z2 = 1 as well, then Vk = 1, k € N will work. Hence we assume that z2 = e'v, (p ^ 0. Let rk be a sequence of numbers with 0 < rk < 1 and rk -> 1 for A: —> oo. We define - rkz2)