Summary: Let \(A\) be the class of analytic functions \(f\left(z \right)\) in the unit disc \(D = \left\{ {z:\left| z \right| < 1} \right\}\) normalized by \(f\left(0 \right) = f'\left(0 \right) - 1 = 0\). A class of generalized analytic functions \(SR_\lambda^*\) on the right-half bounded domain of lemniscate of Bernoulli is introduced, which is shown as follows: \(SR_\lambda^* = \left\{ {f \in A:\left({1 - \lambda} \right)\frac{{f\left(z \right)}}{z} + \lambda f'\left(z \right) \prec \sqrt {1 + z} \left({0 \le \lambda \le 1; z \in D} \right)} \right\}\). And, the third Hankel determinant \({H_3}\left(1 \right)\) for the above function class \(SR_\lambda^*\) is investigated and the upper bound for the above determinant \({H_3}\left(1 \right)\) is obtained.