Third Hankel Determinant for the Inverse of Starlike Functions of Order

Lemma 1). In this way, it was possible to get sharp bounds of H3,1( f ). Similarly, one can obtain estimates of H3,1( f −1). It is worth recalling the most important papers in this direction [2, 4–8, 10, 13, 16, 17]. Turning to S∗(α), the main class of our interest, let us cite the known results. The sharp bounds of H3,1( f ) and H3,1( f −1) in S∗were obtained by Kowalczyk, Lecko, Thomas [6] and by Allu, Thomas (unpublished paper). They proved that |H3,1( f )| ≤4 9 and |H3,1( f −1)| ≤1 for S∗. M

Lemma 1 ( [9, 11, 12]) If p ∈P, then 2p2 = p12 + x(4 −p12) , 4p3 = p13 + 2p1(4 −p12)x −p1(4 −p12)x2 + 2(4 −p12)(1 −|x|2)y , 8p4 = p14 + (4 −p12)x

Theorem 2 If f ∈S∗(α) is given by (1), then |H3,1( f −1)| ≤4 9(1 −α)2 , α ∈  3 8, 17 18  . The result is sharp.

Theorem 2 and Examples 1–3, the following conjecture can be posed |H3,1( f −1)| ≤ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 2(1 −α)2(2 −3α) , α ∈[0, 10 27] 4 9(1 −α)2 , α ∈[ 10 27, 26 27] 1 2(1 −α)2(3α −2) , α ∈[ 26