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Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo form="prefix">sinh</mo><mo>(</mo><mi>z</mi><mo>)</mo><mo>)</mo><mo>≯</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> it implies that the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mo form="prefix">sinh</mo></mrow><mo>*</mo></msubsup></semantics></math></inline-formula> introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with the restriction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>λ</mi><mo>≤</mo><mo form="prefix">ln</mo><mo>(</mo><mn>1</mn><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> and by doing that, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo form="prefix">sinh</mo><mo>(</mo><mi>λ</mi><mi>z</mi><mo>)</mo><mo>)</mo><mo>></mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> The present research intends to provide a novel subclass of starlike functions in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">U</mi><mo>,</mo></mrow></semantics></math></inline-formula> denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mo form="prefix">sinh</mo><mi>λ</mi></mrow><mo>*</mo></msubsup></semantics></math></inline-formula>, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>.</mo></mrow></semantics></math></inline-formula> Then, we prove a lemma, in which the largest disk contained in the image domain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mo form="prefix">sinh</mo><mrow><mo>(</mo><mi>λ</mi><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and the smallest disk containing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>0</mn></msub><mrow><mo>(</mo><mi mathvariant="script">U</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">K</mi><mo>(</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula> of starlike functions of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and convex functions of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. Investigating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mo form="prefix">sinh</mo><mi>λ</mi></mrow><mo>*</mo></msubsup></semantics></math></inline-formula> radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mo form="prefix">sinh</mo><mi>λ</mi></mrow><mo>*</mo></msubsup></semantics></math></inline-formula> radii of different subclasses is the calculation of that value of the radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, is also obtained.