On sharp bounds of certain close-to-convex functions

• Published in: Afrika mathematica Vol. 35; no. 2
• Main Authors: Goel, Priyanka, Kumar, S. Sivaprasad
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2024; Springer Nature B.V
• Subjects: Applications of Mathematics; History of Mathematical Sciences; Mathematics; Mathematics and Statistics; Mathematics Education; Upper bounds; Starlike functions; 30C45; Inverse coefficients; 30C50; Univalent functions; 30C80; Close-to-convex functions
• ISSN: 1012-9405; 2190-7668
• DOI: 10.1007/s13370-024-01176-7

We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities:
Re
(
(
1
-
z
)
f
′
(
z
)
)
>
0
,
Re
(
(
1
-
z
2
)
f
′
(
z
)
)
>
0
,
Re
(
(
1
-
z
+
z
2
)
f
′
(
z
)
)
>
0
and
Re
(
(
1
-
z
)
2
f
′
(
z
)
)
>
0
.