On the difference between permutation polynomials

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d²−3d+4)², then there is no complete mapping polynomial f in F_p[x] of degree d≥2. For arbitrary finite fields F_q, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f. Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over F_p, with p>(d²−3d+4)², then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in F_q[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+x^k is a permutation over F_q, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).