Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

<jats:p>This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>ϕ</mi> </math> </jats:inline-formula>-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>ϕ</mi> </math> </jats:inline-formula>-Laplacian operator on closed oriented <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>m</mi> </math> </jats:inline-formula>-dimensional slant submanifolds in a Sasakian space form <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msup> <mrow> <mover accent="true"> <mi mathvariant="double-struck">M</mi> <mo stretchy="true">~</mo> </mover> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <mi>ε</mi> </mrow> </mfenced> </math> </jats:inline-formula> is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>ϕ</mi> </math> </jats:inline-formula>-Laplacian on slant submanifold in a sphere <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math> </jats:inline-formula> with <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi>ε</mi> <mo>=</mo> <mn>1</mn> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>ϕ</mi> <mo>=</mo> <mn>2</mn> </math> </jats:inline-formula>.</jats:p>

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