| Summary: | Let <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="bold">b</mi><mo>∈</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup><mo>\</mo><mrow><mo>{</mo><mn mathvariant="bold">0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><msup><mi>z</mi><mn>0</mn></msup><mo>+</mo><mi>t</mi><mi mathvariant="bold">b</mi><mo>:</mo><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>}</mo></mrow></semantics></math></inline-formula> with the unit ball <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">B</mi><mi>n</mi></msup><mrow><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo></mrow><msup><mi mathvariant="double-struck">C</mi><mo>:</mo></msup><mrow><mspace width="4pt"></mspace><mo>|</mo><mi>z</mi><mo>|</mo><mo>:</mo></mrow><mo>=</mo><msqrt><mrow><msubsup><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mrow><mo>|</mo><msub><mi>z</mi><mi>n</mi></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></msqrt><mrow><mo><</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> for any <inline-formula><math display="inline"><semantics><mrow><msup><mi>z</mi><mn>0</mn></msup><mo>∈</mo><msup><mi mathvariant="double-struck">B</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>. For this class of functions, there is introduced a concept of boundedness of <i>L</i>-index in the direction <inline-formula><math display="inline"><semantics><mi mathvariant="bold">b</mi></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="bold">L</mi><mo>:</mo><msup><mi mathvariant="double-struck">B</mi><mi>n</mi></msup><mo>→</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> is a positive continuous function such that <inline-formula><math display="inline"><semantics><mrow><mi>L</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>></mo><mfrac><mrow><mi>β</mi><mo>|</mo><mi mathvariant="bold">b</mi><mo>|</mo></mrow><mrow><mn>1</mn><mo>−</mo><mo>|</mo><mi>z</mi><mo>|</mo></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><mi>β</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ’circle’ <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><mi>z</mi><mo>+</mo><mi>t</mi><mi mathvariant="bold">b</mi><mo>:</mo><mo>|</mo><mi>t</mi><mo>|</mo><mo>=</mo><mi>r</mi><mo>/</mo><mi>L</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>}</mo></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>;</mo><mi>β</mi><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>∈</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>. It is estimated by the value of the function at the center of the circle. Other propositions concern a connection between the boundedness of <i>L</i>-index in the direction <inline-formula><math display="inline"><semantics><mi mathvariant="bold">b</mi></semantics></math></inline-formula> of the slice holomorphic function <i>F</i> and the boundedness of <inline-formula><math display="inline"><semantics><msub><mi>l</mi><mi>z</mi></msub></semantics></math></inline-formula>-index of the slice function <inline-formula><math display="inline"><semantics><mrow><msub><mi>g</mi><mi>z</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mi>t</mi><mi mathvariant="bold">b</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><msub><mi>l</mi><mi>z</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>L</mi><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mi>t</mi><mi mathvariant="bold">b</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In addition, we show that every slice holomorphic and joint continuous function in the unit ball has a bounded <i>L</i>-index in direction in any domain compactly embedded in the unit ball and for any continuous function <inline-formula><math display="inline"><semantics><mrow><mi>L</mi><mo>:</mo><msup><mi mathvariant="double-struck">B</mi><mi>n</mi></msup><mo>→</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>.</mo></mrow></semantics></math></inline-formula>
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