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In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>,</mo></mrow></semantics></math></inline-formula> where the functions <i>f</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><msubsup><mi>S</mi><mrow><mo>Σ</mo><mo>,</mo><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>s</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><msubsup><mi>S</mi><mrow><mo>Σ</mo><mo>,</mo><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>s</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><msubsup><mi>S</mi><mrow><mo>Σ</mo><mo>,</mo><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>r</mi></mrow></msubsup><mo>.</mo></mrow></semantics></math></inline-formula> As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefficients for the new function classes.