| Summary: | This study establishes new upper bounds for the mean curvature and constant sectional curvature on Riemannian manifolds for the first positive eigenvalue of the <i>q</i>-Laplacian. In particular, various estimates are provided for the first eigenvalue of the <i>q</i>-Laplace operator on closed orientated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-dimensional special contact slant submanifolds in a Sasakian space form, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mover accent="true"><mi mathvariant="double-struck">M</mi><mo>˜</mo></mover><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, with a constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mn>1</mn></msub></semantics></math></inline-formula>-sectional curvature, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>. From our main results, we recovered the Reilly-type inequalities, which were proven before this study.
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