On the difference of inverse coefficients of convex functions

Lemma 2.1 [2, p. 41] For p 2 P, jcnj  2 for n  1: The inequalities are sharp.

Lemma 2.2 [10] If p 2 P is of the form (2.1) with c1  0, then 2c2 ¼ c2 1 þ ð4  c2 1Þf ð2:2Þ and 4c3 ¼ c3 1 þ ð4  c2 1Þc1fð2  fÞ þ 2ð4  c2 1Þð1  jfj2Þg ð2:3Þ for some f, g 2 D. The next lemma is a special case of more general results due to Choi et al. [12] (see also [13]). Let D :¼ fz 2 C : jzj  1g; and define

Lemma 2.3 [12] If AC  0; then YðA; B; CÞ ¼ jAj þ jBj þ jCj; jBj  2ð1  jCjÞ; 1 þ jAj þ B2 4ð1  jCjÞ ; jBj\2ð1  jCjÞ: 8 < : If AC\0; then YðA; B; CÞ ¼

Lemma 2.4 Let P(t) and Q(t) be (possibly degenerate) real quadratic polynomials. Suppose that PðtÞ [ 0 and QðtÞ [ 0 on an interval I  R and that DP [ 0. If there exists a positive constant T such that (i) DQ  T3=2DP, and (ii) TPðtÞ  QðtÞ for t 2 I, then the function GðtÞ ¼ ffiffiffiffiffiffiffiffi PðtÞ p  ffiffiffiffiffiffiffiffiffi QðtÞ

Theorem 3.1 Let f 2 K and be given by (1.1). Then jA4j  1 24 ð4 þ 21s2 þ s3Þ; if 0  s  2 7 ; sð1  2s2Þ; if 2 7  s  ffiffiffiffiffi 5 59 r ;

Theorem 3.2 Let f 2 Kþ and be given by (1.1). Then jA3 þ A4j  1 3 ð1  sÞð1 þ 2s þ 6s2Þ; 4 7  s  1; 1 24 ð8  16s þ 25s2 þ s3Þ; 0  s  4 7 ; 8 > < >

Proposition 3.3 For a fixed constant c 2 ½1; 2, define Fc : ½0; 1 ½1; 1 ! R by Fcðr; tÞ ¼ 16ð1 þ 6r2 þ r4Þ  4c2ð1 þ 31r2 þ 2r4 þ 24rt þ 10r3tÞ þ c4½36 þ r4 þ 60rt þ 10r3t þ r2ð13 þ 24t2Þ: ð3:17Þ Then Fcðr; tÞ  0 for all ðr; tÞ 2 ½0; 1 ½1; 1.

Proposition 3.4 Define G1 : ½0; 2 ½0; 1 ! R by G1ðc; rÞ ¼ 6c2  5rð4  c2Þ þ ð4  c2Þr2: Then G1ðc; rÞ  0 for all ðc; rÞ 2 ½0; 2 ½0; 1. We are able to now state and prove our main result. 4 Main result

Theorem 4.1 Let f 2 K and be given by (1.1). Then  1 3  jA4j  jA3j  1 4 : ð4:1Þ Both inequalities are sharp.

Theorem 3.1, jA4j  jA3j  jA4j  1=4 when 0  c  2 ffiffiffiffiffiffiffiffiffiffi 5=59 p , and so it is enough to consider c satisfying 2 ffiffiffiffiffiffiffiffiffiffi 5=59 p  c  2. We now use Lemma 2.2, and the fact that jgj  1 to obtain 48ðjA4j  jA3jÞ  j  6c3 þ 5cð4  c2Þf þ cð4  c2Þf2j  j12c2  4ð4  c2Þfj þ 2ð4  c2Þð1  jfj2Þ