Subclasses of Starlike Functions Subordinate to Convex Functions

THEOREM 1. (i) If -1 < B < A ^ 1, then (ii) If a = b, then e*/ /^ c*,Z?2 ~ Q1 + a l S*(a, b) = S*

THEOREM 2. S*(c, d) c S*(a, b) if and only if \a - c\ â b - d and S*[C, D] c S*[A, B] if and only if \AD - BC\ g (A - B) - (C - D).

LEMMA 1. If G is analytic and H is analytic, univalent and convex in A with the range of G'lH' contained in some convex set 3), then the range of numbers (G(z2) - G(z,) )/(H(z2) - H{zx) ) for \zx\ < 1 and \z2\ < 1 is also contained in 3. THEOREM. 3. K(a9 b)cz S*(a9 b) and K[A, B] c S*[A, B].

Lemma 1, z2f'(z2) - zxf\zx) c 2 for \zx\ < 1, \z2\ < 1. f(z2) - f(zx) Setting zx = 0, we have z2f\z2)/f{z2) c 3 for all z2, \z2\ < 1, so that/ e S*(a, b). The proof that K[A, B] c S*[A, B] is identical, with https://doi.org/10.4153/CJM-1985-004-7 Published online by Cambridge University Press

THEOREM 4. With the notation above, (2 - V3 + (a - bf -2 + V3 + (a + bf p(a, b) = mint , I \ + a - b 1 + a + b and TA . J 2(1 - B) - V3(l - Bf + (1 -A)2 p[A, B] = mint 2 - A - B -2(1 + B) + V3(l + B)2 + (1 + v4)2 2 + A + 5 Equality in both cases occurs for f(z)

COROLLARY 1. - 2 + V3 + 4a2 p(a, a) = 1 + 2a for 1/2 < a < 2y /3/3 and p(a, a) = 2 — \/3, the radius of convexity for S, when a ^ 2\/3/3.

COROLLARY 2. 7/"/ e S, then f is convex of order a, 0 = a < I, for 2 - V3 + a2 \z\ < 1 + a Prao/ Since ^[.4, -1] = A"((l - ^)/2), or equivalently /sT[l - 2a, - 1] (2 - V3 + a2 ) p[l - 2a, - 1 ] = minj — , 1 \. v 1 + a J 4. Orders of starlikeness. MacGregor has shown [7] that for/ e AT(«), z in A,

LEMMA 2. Suppose h(z) = 1 + . . . is analytic and convex in A with Re h(z) > 0 for z e A, andp(z) = 1 + . . . is analytic in A with zp'(z) p(z) + ^ -< h{z). If the differential equation q(z) + —•— = h(z) has a univalent solution q(z) in A, then p(z) < q(z) in A.

LEMMA 3. Iff(z) = z + . . . is analytic and convex in A, then so is F^ = ; / o /<'**•

Lemma 2 was proved in [1] and Lemma 3 was proved in [6].

THEOREM 5. Iff e K(a, b) and c = b2 - (1 - a)2, then (zf'/f) < (zF/F) for z G A where F(z) - b \( c + 1 - a LV 1 (eoz - \)/b b , / , 1 - a \

Theorem 5 upon setting a = (1 - AB)/(l - B2) and b = (A - B)/ (1 - B1). Remarks 1. In [1] the corollary was proved for the special case —\^B < 0 and B < A â -B. 2. One could prove this corollary directly by showing that the solution q(z) = zF'(z)/F(z) to the equation 4(z) + z<f{z)/q(z) = (1 + ^z)/(l + £z), - 1 â 5 < v4 â 1 is univalent in A. 3. Since K[A, - 1 ] = ^((1 - A)/2), the result of MacGregor [7] is a special case of the corollary.

THEOREM 6. / ?, Ifl = 1, z + S*(a, b) if and only if for all z in A and all + H 2 f* H (l - zy * o.

THEOREM 7. / e 5*^4, 5] ; / a«a" o«/y if for all z in A and all Ï, l?l = 1, a + K 2 1 ~ a - HZ (1 " zf

LEMMA 4. If<p e K(0) and g e S*(0), then for each function F, analytic in A, the image of A under (<p * Fg)/(y * g) is a subset of the convex hull of F(A).

THEOREM 8. Iff e S*(a, b\ S*[A9 B], K(a, b\ or K[A9 B], then so is f * <pfor any function <p(z) = z -f . . . , analytic and convex in A.

COROLLARY 1. 7/*/ G S*(Û, 6), S*[A, B], K(a, b), or K[A, B], then so 1 + Y fz 7y Jo ty~lf(t)dt, Re y è - 1 / 2 .

COROLLARY 2. Iff G S*(a, b\ S*[A, B], K(a, b) or K[A9 B]9 then so / ; 0 s-xs