Initial logarithmic coefficients for functions starlike with respect to symmetric points

Lemma 1 Let xðzÞ ¼ c1z þ c2z2 þ    be a Schwarz function. Then, for any real numbers l and m such that ðl; mÞ 2 1 2  jlj  2 ; 4 27 ðjlj þ 1Þ3  ðjlj þ 1Þ  m  1 

Lemma 2 Let xðzÞ ¼ c1z þ c2z2 þ    be a Schwarz function. Then jc2j  1  jc1j2 ; jc3j  1  jc1j2  jc2j2 1 þ jc1j ; jc4j  1  jc1j2  jc2j2 :

Lemma 3 Let xðzÞ ¼ c1z þ c2z2 þ    be a Schwarz function. Then jc4 þ 2c1c3 þ c2 2 þ 3c2 1c2 þ c4 1j  1 : Initial logarithmic coefficients for functions starlike... Page 3 of 13 62

Lemma 3 is a particular case of more general theorem which is needed in our proofs. For p in P, the class of analytic functions p such that RepðzÞ [ 0 and pð0Þ ¼ 1, and for w 2 C, Efraimidis in [2] defined the determinant Ak;nðwÞ ¼ pnþk pnþk1 pnþk2 . . . pnþ1 pn wp1 1 0 . . . 0

Theorem 1 If p 2 P and w 2 C, then jAk;nðwÞj  2 maxf1; j1  2wjkg for all integers k  0 and n  1. Applying the correspondence between p 2 P and x 2 B0, pðzÞ ¼ 1 þ xðzÞ 1  xðzÞ ; ð1:6Þ it is possible to obtain the analogous theorem for Schwarz functions. As a corollary, putting w ¼ 0 and k ¼ 3, n ¼ 1, Lemma 3 follows. Consider now the case k þ n ¼ 5 in Theorem 1. Formula (1.5) and (1.6) result in

Corollary 1 If x 2 B0 is of the form (1.4) and l 2 C, jlj  1, then jc5 þ 2lc1c4 þ 2lc2c3 þ 3l2c1c2 2 þ 3l2c2 1c3 þ 4l3c3 1c2 þ l4c5 1j  1 ; ð1:7Þ jc5 þ ð1 þ lÞc1c4 þ 2lc2c3 þ lð1 þ 2lÞc1c2 2 þ lð2 þ lÞc2 1c3 þ l2ð3 þ lÞc3 1c2 þ l3c5 1j  1 ; ð1:8Þ

Theorem 2 If f 2 S S, then jc1j  1 2 ; jc2j  1 2 ; jc3j  1 4 ; jc4j  1 4 ;

Theorem 3 If f 2 KS, then jc1j  1 4 ; jc2j  1 6 ; jc3j  1 16 ; jc4j  13 180 ; jc5j  19