| Summary: | In this note, we consider a subclass <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">H</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of starlike functions <i>f</i> with <inline-formula><math display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>p</mi></mrow></semantics></math></inline-formula> for a prescribed <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Usually, in the study of univalent functions, estimates on the Taylor coefficients, Fekete–Szegö functional or Hankel determinats are given. Another coefficient problem which has attracted considerable attention is to estimate the moduli of successive coefficients <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>|</mo><mo>−</mo><mo>|</mo></mrow><msub><mi>a</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>. Recently, the related functional <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> for the initial successive coefficients has been investigated for several classes of univalent functions. We continue this study and for functions <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><mo>∈</mo><msub><mi mathvariant="script">H</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, we investigate upper bounds of initial coefficients and the difference of moduli of successive coefficients <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>4</mn></msub><mo>−</mo><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>. Estimates of the functionals <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>2</mn></msub><msub><mi>a</mi><mn>4</mn></msub><mo>−</mo><msubsup><mi>a</mi><mn>3</mn><mn>2</mn></msubsup><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>4</mn></msub><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> are also derived. The obtained results expand the scope of the theoretical results related with the functional <inline-formula><math display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula> for various subclasses of univalent functions.
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