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Results & Lemmas (6)
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LEMMA 1.
LEMMA 1. Let w z) be regular in U and such that to(0) = 0. Then if (z) attains its maximum value on the circle — r at a point ZQ 6W, we…
LEMMA 1. Let w{z) be regular in U and such that to(0) = 0. Then if \w(z)\ attains its maximum value on the circle \z\ — r at a point ZQ 6W, we Aave (2.1) W(zo) = kw(z0), where k > 1 is a real number. Applying the above lemma, we derive
THEOREM 1.
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THEOREM 1. If f(z) e A(p) satisfies the condition (1.5), then Therefore, f(z) is p-valently starlike in U. The result is sharp. PROOF: We…
THEOREM 1. If f(z) e A(p) satisfies the condition (1.5), then Therefore, f(z) is p-valently starlike in U. The result is sharp. PROOF: We define the function w{z) by 9 , { ' f{z) ~ Then w(z) is regular in U and iz>(0) = 0. It follows from (2.3) that Suppose that there exists a point ZQ £li such that max \w(z)\ = \w(zo)\ = l.
COROLLARY 1
COROLLARY 1. If f(z) e A(i) satisfies then The result is sharp. Next, we prove
COROLLARY 1 . If f(z) e A(i) satisfies then The result is sharp. Next, we prove
THEOREM 2.
THEOREM 2. If f(z) e A p) satisfies (2-9) then zeu). The result is shaxp. PROOF: Defining the function w(z) by (2.11) f(z) l+w(zY we have…
THEOREM 2. If f(z) e A{p) satisfies (2-9) then {zeu). The result is shaxp. PROOF: Defining the function w(z) by (2.11) f(z) l+w(zY we have zf"(z) p 2zw'(z) (2.12) 1 +
Lemma 1
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Lemma 1 implies that p cos 0 2(1 + cos 6) where iu(zo) = e*fl- This proves that Noting that 5(2) = p1/2/(l + z) is univalent in W and 5(0)…
Lemma 1 implies that p cos 0 2(1 + cos 6) where iu(zo) = e*fl- This proves that Noting that 5(2) = p1/2/(l + z) is univalent in W and 5(0) = p1/2 , so that we see that the result is sharp with the extremal function Setting p = 1 in Theorem 2, we have
COROLLARY 2
COROLLARY 2. If f(z) e A 1) satisfies then REFERENCES [1] I.S. Jack, 'Functions starlike and convex of order a', J. London Math. Soc. 3…
COROLLARY 2 . If f(z) e A{1) satisfies then REFERENCES [1] I.S. Jack, 'Functions starlike and convex of order a', J. London Math. Soc. 3 (1971), 469-474. [2] S.S. Miller and P.T. Mocanu, 'Second order differential inequalities in the complex plane", J. Math. Anal. Appl. 65 (1978), 289-305. [3] M. Nunokawa, 'On the multivalent functions', Indian J. Pure Appl. Math. 20 (1989), 577-582. [4] R. Singh and S. Singh, 'Some sufficient conditions for univalence and starlikeness', Colloq. Math. 47 (1982),
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