| Summary: | In the past few years, many scholars gave much attention to the use of <i>q</i>-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric <i>q</i>-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric <i>q</i>-calculus and the symmetric <i>q</i>-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. First, we illustrate the necessary and sufficient convolution condition for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">S</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula> and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><msubsup><mi mathvariant="script">TS</mi><mrow><mi mathvariant="script">H</mi></mrow><mn>0</mn></msubsup><mo>˜</mo></mover><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi mathvariant="script">A</mi><mo>,</mo><mi mathvariant="script">B</mi></mfenced></mrow></semantics></math></inline-formula>. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of <i>q</i>-starlike and <i>q</i>-convex functions of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results.
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