Close-to-Convexity properties of Clausen's Hypergeometric Function $_3F_2(a,b,c;d,e;z)$

Lemma 5. [7] If An ≥0, {nAn} and {nAn −(n + 1)An+1} both are non-increasing, i.e., {nAn} is monotone of order 2, then f defined by (3) is in S∗.

Lemma 6. [8] Suppose that 1 ≥2A2 ≥· · · ≥nAn ≥· · · ≥0 or 1 ≦2A2 ≤· · · ≤nAn ≤· · · ≤2 Then f(z) is defined by (3) is close-to-convex with respect to −log(1 −z).

Lemma 7. [8] Suppose that f is an odd function ( i.e., the value of A2n in (3) is zero for each n ≥1 ) such that 1 ≥3A3 ≥· · · ≥(2n + 1)A2n+1 ≥· · · ≥0 or 1 ≤3A3 ≤· · · ≤(2n + 1)A2n+1 ≤· · · ≤2 Then f ∈S. In fact, f(z) is close-to-convex with respect to the convex function 1 2 log((1+ z)/(1 −z)). In 1986, Ruscheweyh and Singh [3] obtained the sufficient conditions on the parame- ters a, b and c for z 2F1(a, b; c; z) to be starlike of order β < 1. Further the year 1995, Ponnusamy and Vourinen [4] h

Theorem 8. If a, b, c > 0, de ≥2 a b c and d + e ≥Max  a + b + c, 1 2(ab + bc + ac + 2(a + b + c) −1 −2abc), 2[ab + bc + ac] −3abc  , then z 3F2(a, b, c; d, e; z) is close-to-convex with respect to −log(1 −z).

Theorem 10. If a, b, c > 0, de ≥3abc and d + e ≥Max {a + b + c, α(a, b, c), 3(a b + b c + a c) −7a b c} (11) where α(a, b, c) = 1 3 ((2(a b + b c + a c) + 3(a + b + c) −6 a b c −1) then z 3F2(a, b, c; d, e; z2) is close-to-convex with respect to 1 2 log((1 + z)/(1 −z)).

Theorem 13. Let a, b, and c > 0, d + e ≥Max {T1(a, b, c), T2(a, b, c), T3(a, b, c), T4(a, b, c)}

Theorem 16. Let a, b, and c > 0. Suppose that d + e ≥max{T1(a, b, c), T2(a, b, c), T3(a, b, c)} where T1(a, b, c) = (e + d −c −b −a + 1) (e + d −c −b −a + 2) , T2(a, b, c) = 2 (e + d −c −b −a + 2) (d e + 2 e + 2 d −b c −a c −c −a b −b −a + 1) , T3(a, b, c) = d2 + 7 d + 5
e2 + 7 d2 + ((−2 b −2 a −4) c + (−2 a −4) b −4 a + 21) d