Abstract
In this paper, we define and study a class of analytic functions in the unit disc by
modification of the well-known Robertson’s analytic formula for starlike functions
with respect to a boundary point combined with subordination. An integral repre-
sentation and growth theorem are proved. Early coefficients and the Fekete–Szego¨
functional are also estimated.
Results & Lemmas (11)
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.
Theorem 1
Theorem 1 Let / 2 Pð1Þ and 1 1, A 1: Then Gð/; A; BÞ 6 G: Now we prove the representation theorem which indeed offers a useful…
Theorem 1 Let / 2 Pð1Þ and 1\A 1, A\B\1: Then Gð/; A; BÞ 6 G: Now we prove the representation theorem which indeed offers a useful technique to construct functions in the class Gð/; A; BÞ.
Theorem 2
Theorem 2 Let / 2 Pð1Þ and 1 1, A 1: Then g 2 Gð/; A; BÞ if and only if there exists a function p 2 H such that p / and for z 2 D,…
Theorem 2 Let / 2 Pð1Þ and 1\A 1, A\B 1: Then g 2 Gð/; A; BÞ if and only if there exists a function p 2 H such that p / and for z 2 D, gðzÞ ¼ ð1 BzÞ AþB 2B exp 1 2 Z z 0 pðfÞ 1 f df
Theorem 3
Theorem 3 Let / 2 Pð1Þ and 1 1, A 1 and let 0 1: If g 2 Gð/; A; BÞ; then for B 6¼ 0; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h/ðrÞ r r ð1 jBjrÞ AþB…
Theorem 3 Let / 2 Pð1Þ and 1\A 1, A\B 1 and let 0\r\1: If g 2 Gð/; A; BÞ; then for B 6¼ 0; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h/ðrÞ r r ð1 jBjrÞ AþB 2B jgðzÞj ffiffiffiffiffiffiffiffiffiffiffiffiffiffi h/ðrÞ r r ð1 þ jBjrÞ
Theorem 4
Theorem 4 Let / 2 Pð1Þ and 1 1, A 1 and let 0 1: If g 2 Gð/; A; BÞ; then for B 6¼ 0; arg gðz0Þ ð1 Bz0Þ AþB 2B
Theorem 4 Let / 2 Pð1Þ and 1\A 1, A\B\1 and let 0\r\1: If g 2 Gð/; A; BÞ; then for B 6¼ 0; arg gðz0Þ ð1 Bz0Þ AþB 2B
Lemma 1
Lemma 1 Let / 2 Pð1Þ be a convex function and h 2 A: Then h 2 Sð/Þ if and only if for all jsj 1 and jtj 1, hðszÞ hðtzÞ h/ðszÞ…
Lemma 1 Let / 2 Pð1Þ be a convex function and h 2 A: Then h 2 Sð/Þ if and only if for all jsj 1 and jtj 1, hðszÞ hðtzÞ h/ðszÞ h/ðtzÞ ; z 2 D: ð19Þ 8 Page 8 of 16 A. Lecko et al..
Theorem 5
Theorem 5 Let / 2 Pð1Þ be a convex function and 1 1, A 1: Then g 2 Gð/; A; BÞ if and only if for all jsj 1 and jtj 1, s t 1 …
Theorem 5 Let / 2 Pð1Þ be a convex function and 1\A 1, A\B 1: Then g 2 Gð/; A; BÞ if and only if for all jsj 1 and jtj 1, s t 1 Btz 1 Bsz AþB B gðszÞ gðtzÞ 2 h/ðszÞ
Theorem 6
Theorem 6 Let / 2 Pð1Þ and 1 1, A 1: If g 2 Gð/; A; BÞ, then ðgðzÞÞ2 ð1 zÞ AþB B h/ðzÞ z; z 2 D; B 6¼ 0; ð22Þ and expðAzÞðgðzÞÞ2…
Theorem 6 Let / 2 Pð1Þ and 1\A 1, A\B 1: If g 2 Gð/; A; BÞ, then ðgðzÞÞ2 ð1 zÞ AþB B h/ðzÞ z ; z 2 D; B 6¼ 0; ð22Þ and expðAzÞðgðzÞÞ2 h/ðzÞ z ; z 2 D; B ¼ 0:
Lemma 2
Lemma 2 ([8]) If x 2 B0 is of the form xðzÞ ¼ X 1 n¼1 wnzn; z 2 D; ð24Þ then for m 2 C, jw2 mw2 1j max 1; jmj f g: ð25Þ
Lemma 2 ([8]) If x 2 B0 is of the form xðzÞ ¼ X 1 n¼1 wnzn; z 2 D; ð24Þ then for m 2 C, jw2 mw2 1j max 1; jmj f g: ð25Þ
Lemma 3
Lemma 3 ([17]) If x 2 B; then for any real numbers q1 and q2; the following sharp estimate holds: jw3 þ q1w1w2 þ q2w3 1j Hðq1; q2Þ; ð26Þ…
Lemma 3 ([17]) If x 2 B; then for any real numbers q1 and q2; the following sharp estimate holds: jw3 þ q1w1w2 þ q2w3 1j Hðq1; q2Þ; ð26Þ where Hðq1; q2Þ :¼ 1 for ðq1; q2Þ 2 D1 [ D2; jq2j for ðq1; q2Þ 2 [7 k¼3Dk; 2 3 ðjq1j þ 1Þ
Theorem 7
Theorem 7 Let / 2 Pð1Þ be of the form (1) and 1 1, A 1: If g 2 Gð/; A; BÞ is of the form (2), then j2d1 þ A þ Bj B1; ð27Þ jd1j 1…
Theorem 7 Let / 2 Pð1Þ be of the form (1) and 1\A 1, A\B 1: If g 2 Gð/; A; BÞ is of the form (2), then j2d1 þ A þ Bj B1; ð27Þ jd1j 1 2 ðB1 þ A þ BÞ; ð28Þ 8 Page 10 of 16 A. Lecko et al..
Lemma 4
Lemma 4 ([3]) If x 2 B0 is of the form (24), then jw2 mw2 1j m; m 1; 1; 1 m 1; m; m 1: 8 > < >
Lemma 4 ([3]) If x 2 B0 is of the form (24), then jw2 mw2 1j m; m 1; 1; 1 m 1; m; m 1: 8 > < >
Definitions (1)
Def 1
Definition 1 Let / 2 Pð1Þ and 1 1, A 1: By Gð/; A; BÞ we denote the class of all g 2 H of the form (2) such that 2zg0ðzÞ gðzÞ þ Qðz;…
Definition 1 Let / 2 Pð1Þ and 1\A 1, A\B 1: By Gð/; A; BÞ we denote the class of all g 2 H of the form (2) such that 2zg0ðzÞ gðzÞ þ Qðz; A; BÞ /ðzÞ; z 2 D; ð6Þ where Q is given by (4).
Function classes studied:
Related Papers