<p>This paper is devoted to Wilker–Huygens type inequalities for generalized trigonometric and hyperbolic functions. Building upon earlier work by Neuman, we address several limitations and inaccuracies in the existing literature and present three key advances. First, we correct the flawed condition in Neuman’s work for hyperbolic function inequalities. Second, we extend significantly the parameter range by introducing a refined parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tilde{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> for trigonometric functions and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p^2/(1+2p)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for hyperbolic functions, covering both positive and negative power cases. Third, we adopt a monotonicity analysis of the constructed function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Phi (x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (combined with properties of Gaussian hypergeometric functions) to establish necessary and sufficient conditions for the inequalities, which fills gaps in previous results.</p>