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Abstract
In this paper, we obtain some coefficients bounds (namely, Fekete-Szeg¨o inequalities)
for functions belonging to certain subclass of analytic functions defined by using Salagean op-
erator. Moreover, connections of the results presented here with those obtained in earlier works
are pointed out.
2010 Mathematics Subject Classification: 30C45
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.
Lemma 1. If p.´/ D 1 C c1´ C c2´2 C::: is a function with positive real part in U and is a complex number, then ˇˇc2 c2 1 ˇˇ …
Lemma 1. If p.´/ D 1 C c1´ C c2´2 C ::: is a function with positive real part in U and is a complex number, then ˇˇc2 c2 1 ˇˇ 2maxf1Ij2 1jg: (2.1) The result is sharp for the functions given by p.´/ D 1C´2 1 ´2 and p.´/ D 1C´ 1 ´ .´ 2 U /: (2.2)
Lemma 2.
Lemma 2. If p1.´/ D 1Cc1´Cc2´2 C::: is a function with positive real part in U; then ˇˇc2 c2 1 ˇˇ 8 <: 4 C2; if 0; 2; if 0 …
Lemma 2. If p1.´/ D 1Cc1´Cc2´2 C::: is a function with positive real part in U; then ˇˇc2 c2 1 ˇˇ 8 < : 4 C2; if 0; 2; if 0 1; 4 2; if 1:
Theorem 1.
Theorem 1. Let '.´/ D 1 C B1´ C B2´2 C B3´3 C:::; where '.´/ 2 A and ' 0.0/ > 0: If f.´/ given by (1.1) belongs to the class M b ˛;n.'/ and…
Theorem 1. Let '.´/ D 1 C B1´ C B2´2 C B3´3 C :::; where '.´/ 2 A and ' 0.0/ > 0: If f .´/ given by (1.1) belongs to the class M b ˛;n .'/ and if is a com- plex number, then ˇˇa3 a2 2 ˇˇ jbjB1 2:3n .1C2˛/ max 1; ˇˇˇˇ B2 B1
Theorem 2.
Theorem 2. Let '.´/ D 1CB1´CB2´2CB3´3C:::;.b > 0IBi > 0Ii 2 N/:Also let 1 D 22n 1.1C˛/2 bB2 1 CB2 B1 3n.1C2˛/bB2 1; and 2 D…
Theorem 2. Let '.´/ D 1CB1´CB2´2CB3´3C:::; .b > 0IBi > 0Ii 2 N/:Also let 1 D 22n 1 .1C˛/2 bB2 1 CB2 B1 3n .1C2˛/bB2 1 ; and 2 D 22n 1 .1C˛/2 bB2 1 CB2 CB1 3n .1C2˛/bB2 1 :
Theorem 3.
Theorem 3. For '.´/ D 1CB1´CB2´2 CB3´3 C:::;.b > 0IBi > 0Ii 2 N/ and f.´/ given by (1.1) belongs to the class M b ˛;n.'/ and 1 2;…
Theorem 3. For '.´/ D 1CB1´CB2´2 CB3´3 C:::; .b > 0IBi > 0Ii 2 N/ and f .´/ given by (1.1) belongs to the class M b ˛;n .'/ and 1 2; then in view of
Lemma 2
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Lemma 2, Theorem 2 can be improved. Let 3 D 22n 1.1C˛/2 bB2 1 CB2 3n.1C2˛/bB2 1; (i) If 1 3; then ˇˇa3 a2 2 ˇˇC 22n 1.1C˛/2…
Lemma 2, Theorem 2 can be improved. Let 3 D 22n 1 .1C˛/2 bB2 1 CB2 3n .1C2˛/bB2 1 ; (i) If 1 3; then ˇˇa3 a2 2 ˇˇC 22n 1.1C˛/2 3n.1C2˛/bB2 1 h B1 B2C3n.1C2˛/ 22n 1.1C˛/2