On starlikeness and convexity of certain analytic functions

LEMMA 2.1 [9]. If w(z)e<^, then for \z\ < 1, 1 - |z| 2

LEMMA 2.2. Let wx{z) = [1 - w(z)]/[l + βw(z)]9 where w(z)

LEMMA 2.3. If w(z) e<^,β^0, then for | z \ = r < min (1,1//9), (2.3) Re I 2ί£® 1 < r l[l - w(z)][l + βw(z)] J - (1 - r)(l

THEOREM 3.1. Let f(z) eNbe such that f(z)/[\f(z) + (1 - &ΐ9 where g(z) e N and g(z)/z e&*, 0 ^ λ < ( l + l/3 + l/2Ύ)/(2 + T/T). Γftβ^ ίftβ radius of starlikeness σx of f(z) is given by the only positive root in (0, 1) of the equation βr* + (2 + 3/S)r 2 + 3r - 1 = 0 , where β = [(1 + λ)7 - 1]/(1 - λ)Ύ. Proo/. Put ψ(a ) = 1 - f(z)/7[Xf(z) + (1 - λ)flr(«)]. Then | f{z) \ < 1 f or I»|< 1 and ψ (0) = 1 -1/7 = A. Let t φ ) = [t(») - A]/[l - A ^ ) ] . It is clear that w(z) e έ% and ψ(z) = [^(2;) + -A]/

THEOREM 3.2. Let f(z) eNbe such that f(z)/[xf(z) + (1 -^)g(z)] e <&7, where g(z) e N and g(z)/z e &*1/2. Then the radius of starlίkeness of f(z) is r19 for 0 ^ λ ^ 1/27 , r2 = [2^ 2(1 + /3) 1/2 - 1]/(1 + 2/3) , /or 1/27 < λ < (VT+ 1 + 3) , 8 = [(1 + λ)7 — 1]/(1 — λ)7 and r1 is the smallest positive root in (0, 1) of the equation (1 + 2/3 + 9/3 2)r 4 + 2(1 + 12/3 + 3/3 2)r

THEOREM 3.3. Let f(z)eNbe such that f(z)/[Xf(z) + (1 - X)g(z)] e &r, where g(z) e S*(cκ), 0 ^ X < λ0, some Xo < 1. Tftew the radius of starlikeness σ3 of f(z) is given by the smallest positive root in (0, 1) of the equation β(2a - l)r 3 + (3/5 + 2a - 2aβ)r 2 4- (3 - 2a)r - 1 = 0, where β = [(1 + λ)7 - 1]/(1 - λ)7.

Theorem 3.1 that