| Summary: | Let <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi mathvariant="script">A</mi></mrow></semantics></math></inline-formula>, the class of normalized analytic functions defined in the unit disk <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">D</mi></semantics></math></inline-formula>, and be given by <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></msubsup><msub><mi>a</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>∈</mo><mi mathvariant="double-struck">D</mi></mrow></semantics></math></inline-formula>. This paper presents a new approach to finding bounds for <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>. As an application, we find the sharp bound for <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>5</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> for the class <inline-formula><math display="inline"><semantics><mrow><msub><mi>B</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of Bazilevič functions when <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>.
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