On the univalence of some analytic functions

LEMMA 1. If p(z) = 1 + b^z n + bn+ιzn+ι + is analytic and satisfies Ue(p(z)) > a, 0 ^ α: < 1, /or | s < 1, ίAew (1) p(s) = [1 + {2a - l)z n u(z)]/[l + z nu{z)} ,

LEMMA 3. Under the hypothesis of Lemma 1 we z 11/(1 + | z \ n). /or 13 | < 1 Re

LEMMA 4. ([6]). // h(z) = 1 + cnz n + cn+ίz n+ί + . . . is analytic and Re (h(z)) > 0 for \ z | < 1, then [1 - λ I h(z) I]" 1 ^ (1 - \z /or | z | < [(1 - λ)/(l + λ)] 1/Λ, wfterβ 0 ^ λ < 1. 3.

THEOREM 1. Suppose that f(z) = z + an+1z n+1 + an+2z n+2 H , and g(z) = z + 6 % + 1^ + 1 + bn+2z n+2 + ••• a r e analytic and Re(g(z)/z)>0

COROLLARY 1. Suppose that f(z) = z + an+1z n+1 + an+2z n+2' , αraZ 0(2) = z + 6%+1£ %+1 + 6%+2^ +2 + are analytic and Re (g(z)/z) > 0 /or 2 I < 1. / / R e (f(z)/g{z)) > 0 /or | s | < 1 £/^w /(s) is univalent and starlike for \z\< [(An

THEOREM 2. Suppose f(z) = z + azz 2 + , and

Theorem 2] and improves a result of MacGregor [2, Theorem 4] since Re (g(z)/z) > 1/2 does not necessarily imply that g(z) is convex [7]. The functions f(z) = z(l — z)/(l + z) 2 and g(z) — z/(l + z) satisfy the hypothesis of Theorem 2 with λ = 0 and f(z) is univalent in no circle z I < r with r > 1/3 since f'(z) vanishes at z — 1/3. This shows that Theorem 2 is sharp at least for λ = O A function f(z) = z + ΣΓ=2 a>kZ k is said to be starlike of order a, 0 ^ a < 1, for I 2 I < 1 if Re (zf'(z)/f(z)

THEOREM 3. Let f(z) = z + Σ~=*+i bkz k and g(z) = z + Σ?=«+i ^ ^ δβ analytic for | 2 | < 1 and g{z) be starlike of order a, 0 ^ a < 1, /or I s I < 1. // Re (f(z)/[Xf(z) + (1 - λ)flr(«)]) > 0 /or | s | < 1, is univalent and starlike for ( i ) I z \< [(1 - λ)/(l + λ + 2n)Y !» if a = 1/2 (ϋ) I z\ < R ιln , if a Φ 1/2 , where R = {[A

Theorem 3 reduces to a result of Ratti [5, Theorem 3] The func- tions j\Z) — z{L — z )i\L ~τ z ) a n a gyz) — show that Theorem 3 is sharp at least for λ = 0 and arbitrary n, since the derivative of f(z) vanishes at z - {[(n + 1 - a) - ((n + 1 - a) 2 - (1 - 2a)) ι' 2]/(l - 2a)Y' n for α ^ 1/2 and at z = - l/(2w + 1) when α: = 1/2.

THEOREM 4. Let f(z) = z + an+1z n+1 + α w + 2^ + 2 + •••, and g(z) = « + &%+iZ %+1 + &w+2^ +2 + δβ analytic and satisfy Re (g(z)/z) > 0 /or I s | < 1. / / | f(z)l[Xf(z) + (1 - λ)flr(s)] - 1 | < 1, 0 ^ λ < 1, for | « | < 1, then f(z)eS(R lln), where R is the smallest positive root of the equa-

COROLLARY 2. Suppose f(z) = z + a2z 2 + α32 3 + and #(#) = z + 62£ 2 + δ32 3 + . are analytic and satisfy Re (g(z)/z) > 0 /or I z \< 1. 7/ ]/(z)Mz) - 11< 1 for \z | < 1, ίΛβn | zf(z)/f(z) - 11< 1 /or I 2 | < 1/4(1/17 - 3).

Corollary 2 is sharp.

THEOREM 5. Let f(z) = z + an+1z n+1 + α%+2z w+2 + .. and g{z) = 2 + δw+1£ %+1 + &W+22 W+2 + be analytic for | 2 | < 1 and g(z) be star- like of order a for | z \ < 1, 0 ^ a < 1. 1/ I f(z)/[Xf{z) + (1 - λ)flr(s)] - K l , 0 ^ λ < l , / o r | s | < 1 ,

COROLLARY 3. Suppose f(z) = z + an+1z n+1 + an+2z n+2 + and g(z) = 2 + 6 % + 1^ + 1 + δ%+22 %+2 + are analytic for \z\ <1