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Motivated by the work of [1-3], we study generated elliptic integrals of first and the second kind as an application of eigenfunctions of $p$-Laplacian. Moreover, we establish some identities and sharp inequalities of generalized trigonometric functions, and Grötzsch ring function.

References:
1. D.E. Edmunds, P. Gurka, and J. Lang: Properties of generalized trigonometric functions. J. Approx. Theory V. 164 (1), 2012, 47--56.
2. P. Dr\'abek, and R. Man\'asevich: On the closed solution of p-laplacian nonhomogeneous eigenvalues problems. Diff. and Int. Eqns. 12 (1999), 723--740.
3. P. Lindqvist: Some remarkable sine and cosine functions. Ricerche di Matematica Vol. XLIV (1995), 269--290.