This paper establishes sharp bounds on the parameter \(\beta \in \mathbb {R}\) for differential subordination problems of the form \( p({z}) + \beta {z} p'({z}) \prec {h}({z}) \implies p({z}) \prec \varphi _c({z},\alpha ) = 1 + \alpha {z} e^{z}, \) where \(0 < \alpha \le 1\) , p is analytic in \(\mathbb {D}\) with \(p(0)=1\) , and h(z) is a univalent function mapping \(\mathbb {D}\) onto a convex domain containing 1. We specifically examine important cases including the Bernoulli lemniscate \(\sqrt{1+{z}}\) and exponential function \(e^{z}\) . Our approach leverages geometric properties of Gaussian hypergeometric functions \({_2F_1}(a,b,c;{z})\) and confluent hypergeometric functions \({_1F_1}(a,c;{z})\) to obtain optimal results. The function \(\varphi _c({z},\alpha )\) maps \(\mathbb {D}\) univalently onto a cardioid domain for each \(\alpha \in (0,1]\) , allowing us to introduce and study a new Ma-Minda class of cardioid-starlike functions \(\mathcal {S}^*_c(\alpha )\) . As applications, we derive sufficient conditions for analytic functions to belong to \(\mathcal {S}^*_c(\alpha )\) , extending previous work on geometric function theory. The results are shown to be sharp through both analytical proofs and numerical verification.
