LAUSR

The main aim of this manuscript is to investigate sharp bound on the functional $$|a_{p+1}a_{p+2}-a_{p+3}|$$|ap+1ap+2-ap+3| for functions $$f(z)=z^p+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+a_{p+3}z^{p+3}+\cdots $$f(z)=zp+ap+1zp+1+ap+2zp+2+ap+3zp+3+⋯ belonging to the class $$\mathcal {R}_p(\alpha )$$Rp(α) associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on $$|a_{p+1}a_{p+2}-a_{p+3}|$$|ap+1ap+2-ap+3|, and $$|a_{p+1}a_{p+3}-a_{p+2}^2|$$|ap+1ap+3-ap+22| for functions in the class $$\mathcal {RL}_p(\alpha )$$RLp(α), related to the lemniscate of Bernoulli, are also derived. Further, these estimates are used to derive a bound on the third Hankel determinant.