Addendum to “A generalization of a result on the sum of element orders of a finite group”

Let G be a group of order n and H be a subgroup of order m of G . Denote by ψ H ( G ) the sum of element orders relative to H of G . It is known that if G is nilpotent, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ψ</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>≤</m:mo> <m:msub> <m:mi>ψ</m:mi> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mi>m</m:mi> </m:msub> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:math> ψH(G) ≤ψHₘ(Cₙ) , where H m is the unique subgroup of order m of C n . In this note, we show that this inequality does not hold for infinitely many finite solvable groups.