Geometric properties of new subclasses of analytic functions defined by Opoola differential operator

In this research, we utilize the Opoola differential operator to define new subclasses of starlike and convex functions within the unit disk <span class="math inline">\(U\)</span>: <span class="math inline">\(S^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\)</span>, <span class="math inline">\(K^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\)</span>, <span class="math inline">\(T^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\)</span>, and <span class="math inline">\(C^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\)</span>, characterized by parameters <span class="math inline">\(\alpha\)</span>, <span class="math inline">\(\eta\)</span>, and <span class="math inline">\(\gamma\)</span>, which denote their order and type. We investigate various geometric properties of these functions, including characterization properties, growth and distortion theorems, arithmetic mean, and radius of convexity. The results obtained generalize many existing findings, forming a foundation for further research in the theory of geometric functions. Additionally, we present several corollaries and remarks to illustrate extensions of our results.