Abstract
The object of the present paper 1s to invei;tigatc some argument properties of certain
analytic functions in the open unit disk. 0
I
ur rcsu t contain some 111tcresl111g corollnries as the
special cases.
Results & Lemmas (5)
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Lemma 1
Lemma 1([4]). Leth E C, the class of convex functions in U, and let >.(z) be analytic in U with Rc>.(z)~0. If p(z is analytic in U and p(O)…
Lemma 1([4]). Leth E C, the class of convex functions in U, and let >.(z) be analytic in U with Rc>.(z)~0. If p(z} is analytic in U and p(O) = h(O), then p(z) + >.(z)zp'(z) -< h(z) (z EU) implies p(z) -< h(z) (z E U).
Lemma 2
Lemma 2((5)). Let P(z) be analytic in U, p(O) == I, p(z) f- 0 in U and.rnpposc that there exists a-po-iiifz0 E u· ·s1ich that 1TTJ I…
Lemma 2((5)). Let P(z) be analytic in U, p(O) == I, p(z) f- 0 in U and .rnpposc that there exists a-po-iiifz0 E u· ·s1ich that 1TTJ I argp(z)I < 一for 丨zl < lzol 2 and I argp(zo)I = 7r1] "2 , where 17 > 0. Then we have ZoJJ'(zo) p(zo) = ik17, where
Theorem 2.
Theorem 2. Let f EA and g E S(a, o). ff 丨arg (/3 - ((1 五）鬪臼~))I <言 2: 0,/3 > 1,0 < c5 SI), then I arg (/3 f(z) 7r1] - —)I<- g(z) 2' where…
Theorem 2. Let f EA and g E S(a, o). ff 丨arg (/3 - ((1 五）鬪臼~))I <言 2: 0,/3 > 1,0 < c5 SI), then I arg (/3 f(z) 7r1] - —)I<- g(z) 2' where TJ(O < 1]~I) is the solution of the equaiton (2.1).
Theorem 3.
Theorem 3. Let f E A. If I arg zf, (z) 1ro Jl-m(z)zm 2 I< 一(0 < o~1, m E N), then I arg ( f(z) m 可 -) I<-, z 2 where r,(O < 17~1) is the…
Theorem 3. Let f E A. If I arg zf, (z) 1ro Jl-m(z)zm 2 I< 一(0 < o~1, m E N), then I arg ( f(z) m 可 -) I<-, z 2 where r,(O < 17~1) is the solution of the equation
Theorem 4.
Theorem 4. Let f EA. If I arg zf'(z) 11"0 J1 一tt(z)zu 2 I < - (u > o, o < o s 1), I arg zF,(z) I< 竺 Fl-u(z)zu 2, where F is given by Fu(z)…
Theorem 4. Let f EA. If I arg zf'(z) 11"0 J1 一tt(z)zu 2 I < - (u > o, o < o s 1), I arg zF,(z) I< 竺 Fl-u(z)zu 2 , where F is given by Fu(z) = 7 户一 1 Ju(t)dt
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