Let $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

An adaptive control of multiple robot compliant manipulation of a dynamical environment is synthesized. It may be implemented when: (i) there is no good understanding of all physical effects incorporated in the multiple robots/object/environment system to be controlled; (ii) the parameters of the system are not precisely known, (iii) the system parameters do vary in a known region about their nominal values. The proposed adaptive control law has the inverse dynamics controller structure and is composed of an identification part (parameter update law), and a control law part. It is proved that the proposed adaptive controller is asymptotically stable. The simulation results verify the proposed approach to coordinated adaptive control of multiple robot manipulators in constrained motion tasks. They also verify that the multiple robots/object/environment system can track a step parameter change.