Miskolc Mathematical Notes

Lemma 1 ([10]). Let the function w.´/ 2 ˝ be given by (2.1). Then ˇˇw2 w2 1 ˇˇ 5 maxf1;jjg . 2 C/: The result is sharp for the function given by w.´/ D ´ or w.´/ D ´2 .´ 2 U/:

Lemma 2 ([2], [11]). Let the function w.´/ 2 ˝ be given by (2.1). Then ˇˇw2 w2 1 ˇˇ 5 8 ˆˆˆˆ< ˆˆˆˆ:  . 5 1/ 1 .1 5  5 1/  . = 1/: (2.2) For  < 1 or  > 1; the equality holds true in (2.2) if and only if w.´/ D ´ or one

Lemma 3 ([14]). Let the function w.´/ 2 ˝ be given by (2.1). Then; for any real numbers q1 and q2; the following sharp estimates hold trueW ˇˇw3 Cq1w1w2 Cq2w3 1 ˇˇ 5 H.q1;q2/; (2.3) where H.q1;q2/ D 8 ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ< ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ: 1 .q1;q2/ 2 D1 [D2
jq2j

Theorem 1. Let the function '.´/ be given by '.´/ D 1CB1´CB2´2 C


.B1 > 0/: If the function f .´/ given by .1:1/ belongs to the class S;q;b.p;'/ and  2 C; then ˇˇapC2 a2 pC1 ˇˇ 5 Œpq B1 jbj Œp C2q Œpq

Theorem 2. Let b > 0 and let the function '.´/ be given by '.´/ D 1CB1´CB2´2 C


.Bk > 0I k 2 f1;2g/: If the function f .´/ given by .1:1/ belongs to the class S;q;b.p;'/ and  2 R; then ˇˇapC2 a2 pC1 ˇˇ 5

Theorem 2, we can deduce the corresponding results derived earlier by Ali et al. [2], Aouf et al. [3] and Seoudy and Aouf [19].

Theorem 3. Let the function '.´/ be given by '.´/ D 1CB1´CB2´2 C


.B1 > 0/:

Theorem 4. Let the function '.´/ be given by '.´/ D 1CB1´CB2´2 C


.Bk > 0I k 2 f1;2g/: If the function f .´/ given by .1:1/ belongs to the class S;q;b;ı.p;'/ and  2 R; then ˇˇapC2 a2 pC1 ˇˇ 5

Theorem 4, we can obtain new results for each of the following p-valently analytic function classes: S b;p.'/; Cb;p.'/; S;b;p.'/; Sq;b.'/; Cq;b.'/; S b .'/ and Cb.'/; which are defined in Section 1.

Theorem 4, we can deduce new results for each of the following p-valently analytic