We use the concept of convolution to introduce and study the properties of a unified family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$, $(0\leq\gamma\leq1,\,k\geq0)$, consisting of uniformly $k$-starlike and $k$-convex functions of complex order $b\in\mathbb{C}\setminus\{0\}$ and type $\alpha\in[0,1)$. The family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$ is a generalization of several other families of analytic functions available in literature. Apart from discussing the coefficient bounds, sharp radii estimates, extreme points and the subordination theorem for this family, we settle down the Silverman's conjecture for integral means inequality. Moreover, invariance of this family under certain well-known integral operators is also established in this paper. Some previously known results are obtained as special cases.

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This paper study the two unit warm stand by system in which the demand of items increases arbitrarily for some random amount of duration. Whenever demands of items to which the machines are producing is heavy the standby unit also starts operation and when the demand becomes Normal, the standby unit which is in operation comes into standby mode. Failure of the standby unit remains undetected therefore the standby unit is inspected at random intervals of time. The failure can also be detected at the time of need of standby unit to become operative. If the standby unit is found to be failed in the inspection then it is sent for repair immediately. Failure time distribution for both operative and standby units are assumed to be negative exponential. Regenerative point techniques with markov renewal process is used to obtain various reliability characteristics of system. Repair time distribution of units failed during operation and standby position are same and assumed to be general.

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