Results & Lemmas (3)
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Lemma 2.1.
Lemma 2.1. (Pommerenke [17]). If p ∈P. Then the sharp estimate |ck| ≤2 for each k, (2.2) and ¯¯¯¯¯c2 − c2 1 2 ¯¯¯¯¯ ≤2−|c1|2 2.
Lemma 2.1. (Pommerenke [17]). If p ∈P. Then the sharp estimate |ck| ≤2 for each k, (2.2) and ¯¯¯¯¯c2 − c2 1 2 ¯¯¯¯¯ ≤2−|c1|2 2 .
Lemma 2.2.
Lemma 2.2. (Libera and Zlotkiewicz [12, 13]). Let the function p ∈P be given by the powers series (2.1). Then 2c2 = c2 1 + x(4−c2 1), (2.3)…
Lemma 2.2. (Libera and Zlotkiewicz [12, 13]). Let the function p ∈P be given by the powers series (2.1). Then 2c2 = c2 1 + x(4−c2 1), (2.3) for some x,|x| ≤1, and 4c3 = c3 1 +2(4−c2 1)c1x −c1(4−c2 1)x2 +2(4−c2 1)(1−|x|2)z, (2.4) for some value of z,|z| < 1. 3. Main Result
Theorem 3.3.
Theorem 3.3. Let the function f, given by (1.2) be in the class H n,m λ1,λ2. Then ¯¯a2a4 −a2 3 ¯¯ ≤ 16(1+2λ2)2m 9(n +1)2(n +2)2…
Theorem 3.3. Let the function f , given by (1.2) be in the class H n,m λ1,λ2. Then ¯¯a2a4 −a2 3 ¯¯ ≤ 16(1+2λ2)2m 9(n +1)2(n +2)2 (1+2λ1)2m−2 . The result obtained is sharp.
Function classes studied:
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