Abstract
The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427–1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. However, the correct bound for the latter is stated and proved. Further, a sharp bound on the initial coefficients for normalized analytic function f such that $zf'(z)/f(z)\prec \sqrt{1+\lambda z}$
z
f
′
(
z
)
/
f
(
z
)
≺
1
+
λ
z
, $\lambda \in (0, 1]$
λ
∈
(
0
,
1
]
, have also been obtained, which contain many existing results.
Second Hankel determinant for tilted bi-starlike functions of order β
