And he said to [them] all, If any [man] will come after me, let him deny himself, and take up his cross daily, and follow me.
Note 1 at Lu 9:23: Self-denial is an important part of the Christian life. Jesus sacrificed His life for us, and He demands we die to ourselves that we might experience this new life He has provided. We do this first by recognizing that we can't save ourselves by our own effort and second, by trusting God--not self--for salvation. Then we daily need to deny our own wisdom and seek God's wisdom and direction for our lives.
Self-denial is only good when it is denying ourselves for the singular purpose of exalting Jesus and His will for us in some area of our lives. Some have made a religion out of self-denial and take pride in their denial, not in Jesus' Lordship. This leads to legalism and bondage, which Paul condemned as "will worship" (Col 2:23). We are told not only to deny ourselves but to deny ourselves, take up our cross, and follow Jesus.
The real benefits of fasting come as a result of denying ourselves (see note 1 at Mt 4:2). However, if we fast to glorify ourselves, as the Pharisees did (Mt 6:16 and 23:5), we have no reward from God. All we will get from such a fast is hungry.
Note 2 at Lu 9:23: The cross is what Jesus died on. There are circumstances in our lives that, like the cross of Jesus, give us the opportunity to die to ourselves each day. These are not things like sickness, poverty, etc., for which Jesus' atonement provided redemption (see note 2 at Mt 8:17), but rather things like persecution (from which we are not redeemed, 2Ti 3:12) and the constant battle between our flesh and our born-again spirits (Ga 5:17). Our cross that we must bear is to take God's Word (which is His will) and exalt it above our own will in every situation that we face each day.
Note 3 at Lu 9:23: It is very important that we take up our cross daily. Consistency is one of the most important keys in subduing the flesh. We cannot seek God in "spurts" and expect to reach maturity. The victory goes to those who abide in Him (Ps 91:1; Joh 8:31, and 15:4-7).