| Summary: | In 2019 Seneta has provided a characterization of slowly varying functions <i>L</i> in the Zygmund sense by using the condition, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>)</mo> </mrow> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>. Very recently, we have extended this result by considering a wider class of functions <i>U</i> related to the following more general condition. For each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, for some functions <i>r</i> and <i>g</i>. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.
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