It is well known that <mml:math alttext="$T\left( {r,f} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is differentiable at least for <mml:math alttext="$r &#62; r_0 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>r</mml:mi><mml:mo>&#62;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>. We show that, in fact, <mml:math alttext="$T\left( {r,f} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is differentiable for all but at most one value of <mml:math alttext="$r$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>, and if <mml:math alttext="$T\left( {r,f} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> fails to have a derivative for some value of <mml:math alttext="$r$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>, then <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> is a constant times a quotient of finite Blaschke products.

On the differentiability of <mml:math alttext="$T\left( {r,f} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

On the differentiability of <svg style="vertical-align:-3.56265pt" height="16.6625" version="1.1" viewBox="0 0 48 16.6625" width="48" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g><g transform="matrix(.017,-0,0,-.017,11.384,12.162)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,17.265,12.162)"><path id="x1D45F" d="M393 379q-9 -16 -28 -29q-15 -10 -23 -2q-19 19 -36 19q-21 0 -52 -38q-57 -72 -82 -126l-40 -197q-23 -3 -75 -18l-7 7q49 196 74 335q7 43 -2 43q-7 0 -30 -14.5t-47 -37.5l-16 23q37 42 82 73t67 31q41 0 15 -113l-11 -50h4q41 71 85 117t77 46q29 0 45 -26&#xA;q13 -21 0 -43z" /></g><g transform="matrix(.017,-0,0,-.017,24.439,12.162)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,31.136,12.162)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19&#xA;q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" /></g><g transform="matrix(.017,-0,0,-.017,42.049,12.162)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> 

International Journal of Mathematics and Mathematical Sciences

On the differentiability of T(r,f)

Abstract

Copyright