In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H ₃(1) and H ₂(3) for the well known class 𝓢𝓛^* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: | a 3 2 − a 5 | $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛^* is also estimated. Further, a couple of interesting results of 𝓢𝓛^* are also discussed.

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Let − 1 ≤ B < A ≤ 1 . The condition on β is determined so that 1 + β z p ′ ( z ) / p k ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) ( − 1 < k ≤ 3 ) implies p ( z ) ≺ 1 + z . Similarly, the condition on β is determined so that 1 + β z p ′ ( z ) / p n ( z ) or p ( z ) + β z p ′ ( z ) / p n ( z ) ≺ 1 + z ( n = 0 , 1 , 2 ) implies p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) or 1 + z . In addition to that, the condition on β is derived so that p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) when p ( z ) + β z p ′ ( z ) / p ( z ) ≺ 1 + z . A few more problems of the similar flavor are also considered. MSC:30C80, 30C45.